Fundamental Concepts & Operations on Algebraic Expressions

Explanation

Several parts of an algebraic expression separated by a + or - sign are called the terms of the expression. Multiplication and division do not separate the terms.

An expression containing one term is a monomial, an expression containing two terms is a binomial and an expression containing three terms is a trinomial.

(i) $\frac{1}{2}a^2b^2c^2$ is a product of factors, so it has only one term.

Hence, $\bm{\frac{1}{2}a^2b^2c^2}$ is a monomial.

(ii) $7x \times y^2 \times z^3$ involves only multiplication, so it has only one term.

Hence, $\bm{7x \times y^2 \times z^3}$ is a monomial.

(iii) $\frac{9x^3}{z}$ involves only division, so it has only one term.

Hence, $\bm{\frac{9x^3}{z}}$ is a monomial.

(iv) $2x + 5$ has two terms, namely $2x$ and 5.

Hence, $\bm{2x + 5}$ is a binomial.

(v) $\frac{a}{3} + \frac{b}{6}$ has two terms, namely $\frac{a}{3}$ and $\frac{b}{6}$.

Hence, $\bm{\frac{a}{3} + \frac{b}{6}}$ is a binomial.

(vi) $xy + yz + zx$ has three terms, namely $xy, yz$ and $zx$.

Hence, $\bm{xy + yz + zx}$ is a trinomial.

(vii) $\frac{x^2 - 2y^2 + z^2}{3} = \frac{x^2}{3} - \frac{2y^2}{3} + \frac{z^2}{3}$, which has three terms.

Hence, $\bm{\frac{x^2 - 2y^2 + z^2}{3}}$ is a trinomial.

(viii) $8a \div 9b - 2a^2 \times b^2$ has two terms, namely $8a \div 9b$ and $-2a^2 \times b^2$.

Hence, $\bm{8a \div 9b - 2a^2 \times b^2}$ is a binomial.

(ix) $7x^3 + \frac{2y^2 + 1}{5} = 7x^3 + \frac{2y^2}{5} + \frac{1}{5}$, which has three terms.

Hence, $\bm{7x^3 + \frac{2y^2 + 1}{5}}$ is a trinomial.

Explanation

In a term, the numerical factor attached to the variables is called the numerical coefficient and the non-numeric factor consisting of the variables of the term is called the literal coefficient.

(i) In $-7x^2y$, the numerical factor is -7 and the remaining factor is $x^2y$.

Hence, numerical coefficient = $\bm{-7}$ and literal coefficient = $\bm{x^2y}$.

(ii) In $\pi r^2$, the numerical factor is $\pi$ and the remaining factor is $r^2$.

Hence, numerical coefficient = $\bm{\pi}$ and literal coefficient = $\bm{r^2}$.

(iii) In $-\frac{3}{8}bcx$, the numerical factor is $-\frac{3}{8}$ and the remaining factor is $bcx$.

Hence, numerical coefficient = $\bm{-\frac{3}{8}}$ and literal coefficient = $\bm{bcx}$.

(iv) $\frac{2a}{3} = \frac{2}{3} \times a$, so the numerical factor is $\frac{2}{3}$ and the remaining factor is $a$.

Hence, numerical coefficient = $\bm{\frac{2}{3}}$ and literal coefficient = $\bm{a}$.

(v) $5a^2 \times b \div 2c = \frac{5a^2b}{2c} = \frac{5}{2} \times \frac{a^2b}{c}$, so the numerical factor is $\frac{5}{2}$ and the remaining factor is $\frac{a^2b}{c}$.

Hence, numerical coefficient = $\bm{\frac{5}{2}}$ and literal coefficient = $\bm{\frac{a^2b}{c}}$.

(vi) $-\frac{7pq}{9xy} = -\frac{7}{9} \times \frac{pq}{xy}$, so the numerical factor is $-\frac{7}{9}$ and the remaining factor is $\frac{pq}{xy}$.

Hence, numerical coefficient = $\bm{-\frac{7}{9}}$ and literal coefficient = $\bm{\frac{pq}{xy}}$.

Explanation

Any factor of a term is called the coefficient of the product of all the remaining factors.

(i) Coefficient of $x^2$

$$\begin{aligned}\Rightarrow -\frac{3}{5}x^3y^2z \\[1em] \Rightarrow x^2 \times \Big(-\frac{3}{5}xy^2z\Big)\end{aligned}$$

Hence, the coefficient of $\bm{x^2}$ is $\bm{-\frac{3}{5}xy^2z}$.

(ii) Coefficient of $-yz$

$$\begin{aligned}\Rightarrow -\frac{3}{5}x^3y^2z \\[1em] \Rightarrow (-yz) \times \frac{3}{5}x^3y\end{aligned}$$

Hence, the coefficient of $\bm{-yz}$ is $\bm{\frac{3}{5}x^3y}$.

(iii) Coefficient of $\frac{3}{5}xyz$

$$\begin{aligned}\Rightarrow -\frac{3}{5}x^3y^2z \\[1em] \Rightarrow \frac{3}{5}xyz \times (-x^2y)\end{aligned}$$

Hence, the coefficient of $\bm{\frac{3}{5}xyz}$ is $\bm{-x^2y}$.

(iv) Coefficient of $-x^2y$

$$\begin{aligned}\Rightarrow -\frac{3}{5}x^3y^2z \\[1em] \Rightarrow (-x^2y) \times \frac{3}{5}xyz\end{aligned}$$

Hence, the coefficient of $\bm{-x^2y}$ is $\bm{\frac{3}{5}xyz}$.

Explanation:

Terms having the same literal coefficients are called like terms, otherwise they are called unlike terms.

(i) $\frac{x}{2}, -\frac{x}{3}$

The literal coefficient of each term is $x$. As the literal coefficients are the same, they are like terms.

(ii) $6a^2bc, 6ab^2c$

The literal coefficients are $a^2bc$ and $ab^2c$. As the literal coefficients are different, they are unlike terms.

(iii) $6pq, -3qx$

The literal coefficients are $pq$ and $qx$. As the literal coefficients are different, they are unlike terms.

(iv) $8a^2, -\frac{2}{3}a^2$

The literal coefficient of each term is $a^2$. As the literal coefficients are the same, they are like terms.

(v) $2x, 2y$

The literal coefficients are $x$ and $y$. As the literal coefficients are different, they are unlike terms.

(vi) $3xy^2p, -8py^2x$

The literal coefficient of each term is $pxy^2$, as the order of the variables in a term does not matter. As the literal coefficients are the same, they are like terms.

Hence, the pairs of like terms are (i), (iv) and (vi).

Explanation

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

(i) $1 - x$

The powers of $x$ in the terms are 0 and 1, which are non-negative integers. So, it is a polynomial.

(ii) $3 + y + y^2$

The powers of $y$ in the terms are 0, 1 and 2, which are non-negative integers. So, it is a polynomial.

(iii) $z + \sqrt{z}$

Here $\sqrt{z}$ is $z$ raised to the power $\frac{1}{2}$, which is not a non-negative integer. So, it is not a polynomial.

(iv) $x - \frac{1}{x}$

Here $\frac{1}{x} = x^{-1}$, in which the power of $x$ is -1, which is not a non-negative integer. So, it is not a polynomial.

(v) $x^3 + x\sqrt{x} - x + 2$

Here $x\sqrt{x}$ is $x$ raised to the power $\frac{3}{2}$, which is not a non-negative integer. So, it is not a polynomial.

(vi) $x^2 + y^2 + xy + x^2y^2$

The powers of $x$ and $y$ in each term are non-negative integers. So, it is a polynomial.

(vii) 5

It can be written as $5x^0$, in which the power of $x$ is 0, a non-negative integer. So, it is a polynomial.

(viii) $\frac{1}{3}x^3 - x^4$

The powers of $x$ in the terms are 3 and 4, which are non-negative integers. So, it is a polynomial.

(ix) $x^2 + \sqrt{3}x + 5$

The powers of $x$ in the terms are 2, 1 and 0, which are non-negative integers. Here $\sqrt{3}$ is only a constant coefficient. So, it is a polynomial.

(x) $5x^2 + 6xy - 7\sqrt{y}$

Here $\sqrt{y}$ is $y$ raised to the power $\frac{1}{2}$, which is not a non-negative integer. So, it is not a polynomial.

(xi) $6x^2\sqrt{y} - 3xy + 5$

Here $\sqrt{y}$ is $y$ raised to the power $\frac{1}{2}$, which is not a non-negative integer. So, it is not a polynomial.

Hence, the expressions (i), (ii), (vi), (vii), (viii) and (ix) are polynomials.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $2 - x$, the powers of $x$ in the terms are 0 and 1. The highest power is 1.

Hence, the degree of the polynomial = 1.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $3 - x^2 + x^3$, the powers of $x$ in the terms are 0, 2 and 3. The highest power is 3.

Hence, the degree of the polynomial = 3.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $5x^2 - 6x$, the powers of $x$ in the terms are 2 and 1. The highest power is 2.

Hence, the degree of the polynomial = 2.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $2x^3 - 8x$, the powers of $x$ in the terms are 3 and 1. The highest power is 3.

Hence, the degree of the polynomial = 3.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $1 - x + x^4 - 3x^2$, the powers of $x$ in the terms are 0, 1, 4 and 2. The highest power is 4.

Hence, the degree of the polynomial = 4.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $z^3 - z^4 + 2z^2 - 6$, the powers of $z$ in the terms are 3, 4, 2 and 0. The highest power is 4.

Hence, the degree of the polynomial = 4.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $1 - y - y^2 + 3y^5$, the powers of $y$ in the terms are 0, 1, 2 and 5. The highest power is 5.

Hence, the degree of the polynomial = 5.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $x^2 - \frac{x}{2}$, the powers of $x$ in the terms are 2 and 1. The highest power is 2.

Hence, the degree of the polynomial = 2.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $t^4 - t^3 + 2t - 3t^6$, the powers of $t$ in the terms are 4, 3, 1 and 6. The highest power is 6.

Hence, the degree of the polynomial = 6.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

5 is a constant polynomial, as it can be written as $5x^0$, in which the power of $x$ is 0.

Hence, the degree of the polynomial = 0.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $9 - x^2$, the powers of $x$ in the terms are 0 and 2. The highest power is 2.

Hence, the degree of the polynomial = 2.

The highest power of the variable in a polynomial of one variable is called the degree of the polynomial.

In $1 - x^3$, the powers of $x$ in the terms are 0 and 3. The highest power is 3.

Hence, the degree of the polynomial = 3.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $xy = 1 + 1 = 2$

Sum of powers of $yz = 1 + 1 = 2$

Sum of powers of $zx = 1 + 1 = 2$

Sum of powers of $3xyz = 1 + 1 + 1 = 3$

The greatest sum is 3.

Hence, the degree of the polynomial = 3.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $a^2 = 2$

Sum of powers of $b^2 = 2$

Sum of powers of $c^2 = 2$

Sum of powers of $-3abc = 1 + 1 + 1 = 3$

The greatest sum is 3.

Hence, the degree of the polynomial = 3.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $2xy = 1 + 1 = 2$

Sum of powers of $3xy^2 = 1 + 2 = 3$

Sum of powers of $5x^2y = 2 + 1 = 3$

Sum of powers of $7x^2y^2 = 2 + 2 = 4$

The greatest sum is 4.

Hence, the degree of the polynomial = 4.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $a^5 = 5$

Sum of powers of $-b^5 = 5$

Sum of powers of $-2a^3b^3 = 3 + 3 = 6$

The greatest sum is 6.

Hence, the degree of the polynomial = 6.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $x^2y = 2 + 1 = 3$

Sum of powers of $xy^2 = 1 + 2 = 3$

Sum of powers of $5xy = 1 + 1 = 2$

The greatest sum is 3.

Hence, the degree of the polynomial = 3.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of 1 = 0

Sum of powers of $2x = 1$

Sum of powers of $5x^2y = 2 + 1 = 3$

Sum of powers of $6yz^2 = 1 + 2 = 3$

The greatest sum is 3.

Hence, the degree of the polynomial = 3.

Given,

Expression: $4x^3 - 3x^2 + 5x - 6$

Substituting $x = 3$ in the given expression, we get :

$$\begin{aligned}\Rightarrow 4(3)^3 - 3(3)^2 + 5(3) - 6 \\[1em] \Rightarrow 4 \times 27 - 3 \times 9 + 15 - 6 \\[1em] \Rightarrow 108 - 27 + 15 - 6 \\[1em] \Rightarrow 123 - 33 \\[1em] \Rightarrow 90\end{aligned}$$

Hence, the value of the expression = 90.

Given,

Expression: $x^3 - 8x^2 + 14x - 7$

Substituting $x = -1$ in the given expression, we get :

$$\begin{aligned}\Rightarrow (-1)^3 - 8(-1)^2 + 14(-1) - 7 \\[1em] \Rightarrow -1 - 8 \times 1 - 14 - 7 \\[1em] \Rightarrow -1 - 8 - 14 - 7 \\[1em] \Rightarrow -30\end{aligned}$$

Hence, the value of the expression = -30.

Given,

Expression: $a^3 + b^3 - 3a^2b + 3ab^2$

Substituting $a = 4$ and $b = 5$ in the given expression, we get :

$$\begin{aligned}\Rightarrow a^3 + b^3 - 3a^2b + 3ab^2 \\[1em] \Rightarrow (4)^3 + (5)^3 - 3(4)^2(5) + 3(4)(5)^2 \\[1em] \Rightarrow 64 + 125 - 3 \times 16 \times 5 + 3 \times 4 \times 25 \\[1em] \Rightarrow 64 + 125 - 240 + 300 \\[1em] \Rightarrow 489 - 240 \\[1em] \Rightarrow 249\end{aligned}$$

Hence, the value of the expression = 249.

Substituting $x = 4, y = 3$ and $z = -2$ in the given expression, we get :

$$\begin{aligned}\Rightarrow x^2 + y^2 + z^2 + 2xyz \\[1em] \Rightarrow (4)^2 + (3)^2 + (-2)^2 + 2(4)(3)(-2) \\[1em] \Rightarrow 16 + 9 + 4 - 48 \\[1em] \Rightarrow 29 - 48 \\[1em] \Rightarrow -19\end{aligned}$$

Hence, the value of the expression = -19.

Substituting $x = 4, y = 3$ and $z = -2$ in the given expression, we get :

$$\begin{aligned}\Rightarrow x^3 + y^3 + z^3 - 3xyz \\[1em] \Rightarrow (4)^3 + (3)^3 + (-2)^3 - 3(4)(3)(-2) \\[1em] \Rightarrow 64 + 27 - 8 + 72 \\[1em] \Rightarrow 163 - 8 \\[1em] \Rightarrow 155\end{aligned}$$

Hence, the value of the expression = 155.

(i) $7x + y$ is the same as $7xy$

False.

Reason — $7x + y$ is a binomial having two terms $7x$ and $y$, while $7xy$ is a monomial having only one term.

Hence, the correct statement is $\bm{7x + y}$ ≠ $\bm{7xy}$.

(ii) $3x \times 2p \div 5q$ is a trinomial

False.

Reason — Multiplication and division do not separate the terms. So, $3x \times 2p \div 5q = \frac{6px}{5q}$, which has only one term.

Hence, the correct statement is $\bm{3x \times 2p \div 5q}$ is a monomial.

(iii) $2xy + 3yx + 8yz$ is a binomial

True.

(iv) $8p^2q^3r = 8pq^2r^3$

False.

Reason — The powers of the variables in the two terms are different, so they are unlike terms.

Hence, the correct statement is $\bm{8p^2q^3r}$ ≠ $\bm{8pq^2r^3}$.

(v) $3x^2y + 5yx^2$ is a monomial

True.

(vi) $9p \times 6q + 5r$ is a trinomial

False.

Reason — Multiplication does not separate the terms. So, $9p \times 6q + 5r = 54pq + 5r$, which has two terms.

Hence, the correct statement is $\bm{9p \times 6q + 5r}$ is a binomial.

(vii) $-3ab^2c + 4cab^2 + 9b^2ac$ is a trinomial

False.

Reason — All three terms have the same literal coefficient $ab^2c$, so they are like terms and can be added. So, $-3ab^2c + 4cab^2 + 9b^2ac = (-3 + 4 + 9)ab^2c = 10ab^2c$, which has only one term.

Hence, the correct statement is $\bm{-3ab^2c + 4cab^2 + 9b^2ac}$ is a monomial.

(viii) $6x^2yz + 5xy^2z - 2xyz^2$ is a trinomial

True.

(ix) $-7pq^2 + 7p^2q = 0$

False.

Reason — $-7pq^2$ and $7p^2q$ are unlike terms, as the powers of the variables in them are different. So, they cannot be added.

Hence, the correct statement is $\bm{-7pq^2 + 7p^2q}$ ≠ $\bm{0}$

(x) $9x \div 8y$ is a binomial

False.

Reason — Division does not separate the terms. So, $9x \div 8y = \frac{9x}{8y}$, which has only one term.

Hence, the correct statement is $\bm{9x \div 8y}$ is a monomial.

Adding:

$$\begin{aligned}\Rightarrow 2ab + (-7ab) + \frac{2}{3}ab + 9ab \\[1em] \Rightarrow \Big[2 + (-7) + \frac{2}{3} + 9\Big]ab \\[1em] \Rightarrow \Big(4 + \frac{2}{3}\Big)ab \\[1em] \Rightarrow \frac{12 + 2}{3}ab \\[1em] \Rightarrow \frac{14}{3}ab\end{aligned}$$

Hence, the required sum = $\bm{\frac{14}{3}ab}$.

Explanation:

The sum of two or more like terms is the like term whose numerical coefficient is the sum of the numerical coefficients of the given terms.

Adding:

$$\begin{aligned}\Rightarrow 5x^2y + (-3x^2y) + \frac{1}{2}x^2y + \frac{4}{5}x^2y \\[1em] \Rightarrow \Big[5 + (-3) + \frac{1}{2} + \frac{4}{5}\Big]x^2y \\[1em] \Rightarrow \Big(2 + \frac{1}{2} + \frac{4}{5}\Big)x^2y \\[1em] \Rightarrow \frac{20 + 5 + 8}{10}x^2y \\[1em] \Rightarrow \frac{33}{10}x^2y\end{aligned}$$

Hence, the required sum = $\bm{\frac{33}{10}x^2y}$.

Explanation:

The sum of two or more like terms is the like term whose numerical coefficient is the sum of the numerical coefficients of the given terms.

Subtracting:

$$\begin{aligned}\Rightarrow 2xy - 7xy \\[1em] \Rightarrow (2 - 7)xy \\[1em] \Rightarrow -5xy\end{aligned}$$

Hence, the required difference = $\bm{-5xy}$.

Explanation:

The difference of two like terms is the like term whose numerical coefficient is the difference of the numerical coefficients of the given terms.

Subtracting:

$$\begin{aligned}\Rightarrow -3ab - 2ab \\[1em] \Rightarrow (-3 - 2)ab \\[1em] \Rightarrow -5ab\end{aligned}$$

Hence, the required difference = $\bm{-5ab}$.

Explanation:

The difference of two like terms is the like term whose numerical coefficient is the difference of the numerical coefficients of the given terms.

Subtracting:

$$\begin{aligned}\Rightarrow 2a^2b - (-3a^2b) \\[1em] \Rightarrow [2 - (-3)]a^2b \\[1em] \Rightarrow (2 + 3)a^2b \\[1em] \Rightarrow 5a^2b\end{aligned}$$

Hence, the required difference = $\bm{5a^2b}$.

Explanation:

The difference of two like terms is the like term whose numerical coefficient is the difference of the numerical coefficients of the given terms.

Subtracting:

$$\begin{aligned}\Rightarrow -3xy^2 - (-5xy^2) \\[1em] \Rightarrow [-3 - (-5)]xy^2 \\[1em] \Rightarrow (-3 + 5)xy^2 \\[1em] \Rightarrow 2xy^2\end{aligned}$$

Hence, the required difference = $\bm{2xy^2}$.

Explanation:

The difference of two like terms is the like term whose numerical coefficient is the difference of the numerical coefficients of the given terms.

Subtracting:

$$\begin{aligned}\Rightarrow \frac{1}{3}x^2y - \Big(-\frac{3}{7}x^2y\Big) \\[1em] \Rightarrow \Big(\frac{1}{3} + \frac{3}{7}\Big)x^2y \\[1em] \Rightarrow \frac{7 + 9}{21}x^2y \\[1em] \Rightarrow \frac{16}{21}x^2y\end{aligned}$$

Hence, the required difference = $\bm{\frac{16}{21}x^2y}$.

Explanation:

The difference of two like terms is the like term whose numerical coefficient is the difference of the numerical coefficients of the given terms.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrr} & 3x & +8y & -5z \cr & 7x & -9y & +6z \cr & x & -2y & -3z \cr \mathord{+} & -5x & +3y & +z \cr \hline & 6x & & -z \end{array}$$

Hence, the required sum = $\bm{6x - z}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrr} & - x^2 & + y^2 \cr & 2x^2 & - 3y^2 \cr & - 3x^2 & + 5y^2 \cr \mathord{+} & 6x^2 & + 2y^2 \cr \hline & 4x^2 & + 5y^2 \end{array}$$

Hence, the required sum = $\bm{4x^2 + 5y^2}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrr} & 2p & + 4q & + 3r \cr & 3p & - 2q & + 7r \cr & - p & + q & - r \cr \mathord{+} & 5p & + 4q & - 8r \cr \hline & 9p & + 7q & + r \end{array}$$

Hence, the required sum = $\bm{9p + 7q + r}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrr} & 7xy & + yz & - 3zx \cr & - 5xy & + 2yz & + zx \cr \mathord{+} & 4xy & - 3yz & + 2zx \cr \hline & 6xy & & \end{array}$$

Hence, the required sum = $\bm{6xy}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrrr} & - 3x^3 & - x^2 & - x & + 1 \cr & x^3 & + 2x^2 & & + 3 \cr & & x^2 & + 5x & - 2 \cr \mathord{+} & x^3 & - x^2 & - 3x & \cr \hline & - x^3 & + x^2 & + x & + 2 \end{array}$$

Hence, the required sum = $\bm{-x^3 + x^2 + x + 2}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrrrr} & 4z^4 & - 7z^3 & - 5z^2 & + 2z & + 3 \cr & - 2z^4 & + 4z^3 & + 3z^2 & - z & - 5 \cr \mathord{+} & & 3z^3 & + z^2 & - z & - 1 \cr \hline & 2z^4 & & - z^2 & & - 3 \end{array}$$

Hence, the required sum = $\bm{2z^4 - z^2 - 3}$.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrrr} & 7y^3 & - 4y^2 & + 5y & + 3 \cr & 3y^3 & & + 2y & - 7 \cr \mathord{+} & - 9y^3 & + 2y^2 & - 6y & + 5 \cr \hline & y^3 & - 2y^2 & + y & + 1 \end{array}$$

Hence, the required sum = $\bm{y^3 - 2y^2 + y + 1}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrl} & - 4y & + 6z & \\ & 3y & - 5z & \\ &- &+ & \\ \hline & - 7y & + 11z & \\ \end{array}$$

Hence, the required difference = $\bm{-7y + 11z}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrl} & 2a & - 3b & - c & \\ & 3a & - 5b & - 2c & \\ &- &+ &+ & \\ \hline & - a & + 2b & + c & \\ \end{array}$$

Hence, the required difference = $\bm{-a + 2b + c}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrl} & - p^2 & + q^2 & - 2r^2 & \\ & p^2 & - q^2 & + 5r^2 & \\ &- &+ &- & \\ \hline & - 2p^2 & + 2q^2 & - 7r^2 & \\ \end{array}$$

Hence, the required difference = $\bm{-2p^2 + 2q^2 - 7r^2}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrl} & x^2 & - 2x & + 8 & \\ & 5x^2 & + x & - 11 & \\ &- &- &+ & \\ \hline & - 4x^2 & - 3x & + 19 & \\ \end{array}$$

Hence, the required difference = $\bm{-4x^2 - 3x + 19}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrrl} & 2x^3 & - x^2 & + 3x & + 2 & \\ & - x^3 & + 2x^2 & - x & + 1 & \\ &+ &- &+ &- & \quad \\ \hline & 3x^3 & - 3x^2 & + 4x & + 1 & \\ \end{array}$$

Hence, the required difference = $\bm{3x^3 - 3x^2 + 4x + 1}$.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrrl} & - 2x^3 & + 3x^2y & - 5xy^2 & + y^3 & \\ & 7x^3 & - 6x^2y & + 9xy^2 & - 2y^3 & \\ &- &+ &- &+ & \\ \hline & - 9x^3 & + 9x^2y & - 14xy^2 & + 3y^3 & \\ \end{array}$$

Hence, the required difference = $\bm{-9x^3 + 9x^2y - 14xy^2 + 3y^3}$.

First we find the sum of the two given polynomials.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrrr} & 4x^2 & + 7xy & + 3y^2 & + 1 \cr \mathord{+} & 2x^2 & - 5xy & - 2y^2 & + 8 \cr \hline & 6x^2 & + 2xy & + y^2 & + 9 \end{array}$$

Now we subtract this sum from $9x^2 - 8xy + 11y^2$.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrrl} & 9x^2 & - 8xy & + 11y^2 & & \cr & 6x^2 & + 2xy & + y^2 & + 9 & \cr &- &- &- &- & \cr \hline & 3x^2 & - 10xy & + 10y^2 & - 9 & \end{array}$$

Hence, the required difference = $\bm{3x^2 - 10xy + 10y^2 - 9}$.

Let the required expression be $A$.

Then, $2x^2 + 6x - 5 + A = 3x^2 - 2x + 6$

$$\begin{aligned}\Rightarrow A = 3x^2 - 2x + 6 - (2x^2 + 6x - 5) \\[1em] \Rightarrow A = 3x^2 - 2x + 6 - 2x^2 - 6x + 5 \\[1em] \Rightarrow A = (3x^2 - 2x^2) + (-2x - 6x) + (6 + 5) \\[1em] \Rightarrow A = x^2 - 8x + 11\end{aligned}$$

Hence, $\bm{x^2 - 8x + 11}$ must be added.

Let the required expression be $A$.

Then, $4 - 3x + 9x^2 + A = 3 - 4x^2 - x^3$

$$\begin{aligned}\Rightarrow A = 3 - 4x^2 - x^3 - (4 - 3x + 9x^2) \\[1em] \Rightarrow A = 3 - 4x^2 - x^3 - 4 + 3x - 9x^2 \\[1em] \Rightarrow A = -x^3 + (-4x^2 - 9x^2) + 3x + (3 - 4) \\[1em] \Rightarrow A = -x^3 - 13x^2 + 3x - 1\end{aligned}$$

Hence, $\bm{-x^3 - 13x^2 + 3x - 1}$ must be added.

Let the required expression be $A$.

Then, $3x^2y - 2xy^2 + 7x - 2y - A = 7xy^2 - 5x^2y - 3x + 3y$

$$\begin{aligned}\Rightarrow A = 3x^2y - 2xy^2 + 7x - 2y - (7xy^2 - 5x^2y - 3x + 3y) \\[1em] \Rightarrow A = 3x^2y - 2xy^2 + 7x - 2y - 7xy^2 + 5x^2y + 3x - 3y \\[1em] \Rightarrow A = (3x^2y + 5x^2y) + (-2xy^2 - 7xy^2) + (7x + 3x) + (-2y - 3y) \\[1em] \Rightarrow A = 8x^2y - 9xy^2 + 10x - 5y\end{aligned}$$

Hence, $\bm{8x^2y - 9xy^2 + 10x - 5y}$ must be subtracted.

Let the required expression be $A$.

Then, $a^2 + b^2 + c^2 - 3abc - A = 2a^2 - b^2 - 3c^2 + abc$

$$\begin{aligned}\Rightarrow A = a^2 + b^2 + c^2 - 3abc - (2a^2 - b^2 - 3c^2 + abc) \\[1em] \Rightarrow A = a^2 + b^2 + c^2 - 3abc - 2a^2 + b^2 + 3c^2 - abc \\[1em] \Rightarrow A = (a^2 - 2a^2) + (b^2 + b^2) + (c^2 + 3c^2) + (-3abc - abc) \\[1em] \Rightarrow A = -a^2 + 2b^2 + 4c^2 - 4abc\end{aligned}$$

Hence, $\bm{-a^2 + 2b^2 + 4c^2 - 4abc}$ must be subtracted.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrrl} & & & b^3 & - 3b^2 & \\ & a^3 & - 3a^2 & & & \\ &- &+ & & & \\ \hline & - a^3 & + 3a^2 & + b^3 & - 3b^2 & \\ \end{array}$$

Hence, the required difference = $\bm{-a^3 + 3a^2 + b^3 - 3b^2}$.

As the perimeter of a quadrilateral is the sum of the lengths of its four sides,

Perimeter = $(2x - 3y) + (7x + y) + (x + 12y) + (3y - 4x)$

Adding the four expressions by the column method,

$$\begin{array}{rrr} & 2x & - 3y \cr & 7x & + y \cr & x & + 12y \cr \mathord{+} & - 4x & + 3y \cr \hline & 6x & + 13y \end{array}$$

Hence, the perimeter of the quadrilateral = $\bm{(6x + 13y)}$ units.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow 8x^2 \times (-5x) \\[1em] \Rightarrow [8 \times (-5)] \times (x^2 \times x) \\[1em] \Rightarrow -40x^3\end{aligned}$$

Hence, the required product = $\bm{-40x^3}$.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow \frac{2}{3}ab \times (-6a^2b) \\[1em] \Rightarrow \Big[\frac{2}{3} \times (-6)\Big] \times (ab \times a^2b) \\[1em] \Rightarrow -4a^3b^2\end{aligned}$$

Hence, the required product = $\bm{-4a^3b^2}$.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow 7x^2y^3 \times (-4x^3y) \\[1em] \Rightarrow [7 \times (-4)] \times (x^2y^3 \times x^3y) \\[1em] \Rightarrow -28x^5y^4\end{aligned}$$

Hence, the required product = $\bm{-28x^5y^4}$.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow \Big(-\frac{5}{9}ax^2\Big) \times \Big(-\frac{3}{5}bxa^2\Big) \\[1em] \Rightarrow \Big[\Big(-\frac{5}{9}\Big) \times \Big(-\frac{3}{5}\Big)\Big] \times (ax^2 \times bxa^2) \\[1em] \Rightarrow \frac{1}{3}a^3bx^3\end{aligned}$$

Hence, the required product = $\bm{\frac{1}{3}a^3bx^3}$.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow \Big(-\frac{2}{3}a^6b^5\Big) \times \Big(-\frac{9}{4}a^2b^3\Big) \\[1em] \Rightarrow \Big[\Big(-\frac{2}{3}\Big) \times \Big(-\frac{9}{4}\Big)\Big] \times (a^6b^5 \times a^2b^3) \\[1em] \Rightarrow \frac{3}{2}a^8b^8\end{aligned}$$

Hence, the required product = $\bm{\frac{3}{2}a^8b^8}$.

Product of monomials = (Product of their numerical coefficients) × (Product of their variable parts)

$$\begin{aligned}\Rightarrow \Big(-\frac{5}{8}p^2q\Big) \times \frac{16}{25}pq^2 \\[1em] \Rightarrow \Big[\Big(-\frac{5}{8}\Big) \times \frac{16}{25}\Big] \times (p^2q \times pq^2) \\[1em] \Rightarrow -\frac{2}{5}p^3q^3\end{aligned}$$

Hence, the required product = $\bm{-\frac{2}{5}p^3q^3}$.

In this case, we apply distributive laws and thus, multiply each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow (2x^2 - 5x + 6) \times (-3x) \\[1em] \Rightarrow [2x^2 \times (-3x)] - [5x \times (-3x)] + [6 \times (-3x)] \\[1em] \Rightarrow -6x^3 + 15x^2 - 18x\end{aligned}$$

Hence, the required product = $\bm{-6x^3 + 15x^2 - 18x}$.

In this case, we apply distributive laws and thus, multiply each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow (3 - 2y - y^2) \times (-4xy) \\[1em] \Rightarrow [3 \times (-4xy)] - [2y \times (-4xy)] - [y^2 \times (-4xy)] \\[1em] \Rightarrow -12xy + 8xy^2 + 4xy^3\end{aligned}$$

Hence, the required product = $\bm{-12xy + 8xy^2 + 4xy^3}$.

In this case, we apply distributive laws and thus, multiply each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow (3x^2y - 2xy^2 + 5xy - 6) \times 4xy \\[1em] \Rightarrow (3x^2y \times 4xy) - (2xy^2 \times 4xy) + (5xy \times 4xy) - (6 \times 4xy) \\[1em] \Rightarrow 12x^3y^2 - 8x^2y^3 + 20x^2y^2 - 24xy\end{aligned}$$

Hence, the required product = $\bm{12x^3y^2 - 8x^2y^3 + 20x^2y^2 - 24xy}$.

In this case, we apply distributive laws and thus, multiply each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow (7a^3 - 5ab^2 - 2b^3 + 3ab + 2a - 5) \times (-3ab^2) \\[1em] \Rightarrow [7a^3 \times (-3ab^2)] - [5ab^2 \times (-3ab^2)] - [2b^3 \times (-3ab^2)] \\[1em] \quad + : [3ab \times (-3ab^2)] + [2a \times (-3ab^2)] - [5 \times (-3ab^2)] \\[1em] \Rightarrow -21a^4b^2 + 15a^2b^4 + 6ab^5 - 9a^2b^3 - 6a^2b^2 + 15ab^2\end{aligned}$$

Hence, the required product = $\bm{-21a^4b^2 + 15a^2b^4 + 6ab^5 - 9a^2b^3 - 6a^2b^2 + 15ab^2}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{a^2 + 9a + 20}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{x^2 + 5x - 24}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{y^2 - 10y + 24}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{z^2 - 6z - 7}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{6a^2 + 5ab - 6b^2}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{10x^2 - 7xy - 12y^2}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{8x^3 + 18x^2 - x - 15}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Arranging the multiplicand in descending powers of $x$, we get $5x^2 - 2x + 3$.

Hence, the required product = $\bm{25x^3 - 30x^2 + 23x - 12}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{2x^4 + 9x^3 - 10x^2 - 39x + 27}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{8x^3 - 10x^2y + 15xy^2 - 27y^3}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{8x^4 - 4x^3 - 10x^2 + 9x - 2}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{12x^4 - 5x^3 - 42x^2 + 43x - 10}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{4x^4 + 20x^3y - 17x^2y^2 + 21xy^3 + 3y^4}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{6x^5 + 37x^4 - 37x^3 + 34x^2 + 3x - 1}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{16x^6 - 40x^5 + 18x^4 + 33x^3 - 70x^2 + 67x - 24}$.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging the multiplicand in descending order of powers and then multiplying, we get :

Hence, the required product = $\bm{3x^8 - 6x^7 + 2x^6 + 13x^5 - 26x^4 + 19x^3 - 13x^2 + 13x - 4}$.

Number of shelves = $(3x + 5y)$

Number of books on each shelf = $(5x + 3y)$

Total number of books = Number of shelves × Number of books on each shelf

Hence, the number of books in the library = $\bm{15x^2 + 34xy + 15y^2}$.

Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)

$$\begin{aligned}\Rightarrow 45x^7 \div (-9x^4) \\[1em] \Rightarrow \frac{45x^7}{-9x^4} \\[1em] \Rightarrow -\frac{45}{9} \times \frac{x^7}{x^4} \\[1em] \Rightarrow -5x^{(7-4)} \\[1em] \Rightarrow -5x^3\end{aligned}$$

Hence, the required quotient = $\bm{-5x^3}$.

Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)

$$\begin{aligned}\Rightarrow -60x^3y^2 \div (-15xy) \\[1em] \Rightarrow \frac{-60x^3y^2}{-15xy} \\[1em] \Rightarrow \frac{60}{15} \times \frac{x^3y^2}{xy} \\[1em] \Rightarrow 4x^{(3-1)}y^{(2-1)} \\[1em] \Rightarrow 4x^2y\end{aligned}$$

Hence, the required quotient = $\bm{4x^2y}$.

Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)

$$\begin{aligned}\Rightarrow -\frac{3}{4}x^2yz^3 \div \Big(-\frac{2}{3}x^2yz\Big) \\[1em] \Rightarrow \frac{-\frac{3}{4}x^2yz^3}{-\frac{2}{3}x^2yz} \\[1em] \Rightarrow \Big(\frac{3}{4} \times \frac{3}{2}\Big) \times \frac{x^2yz^3}{x^2yz} \\[1em] \Rightarrow \frac{9}{8} \times (x^{(2-2)} \times y^{(1-1)} \times z^{(3-1)}) \\[1em] \Rightarrow \frac{9}{8}z^2\end{aligned}$$

Hence, the required quotient = $\bm{\frac{9}{8}z^2}$.

Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)

$$\begin{aligned}\Rightarrow 63a^4b^3c^6 \div (-14a^2b^5c^4) \\[1em] \Rightarrow \frac{63a^4b^3c^6}{-14a^2b^5c^4} \\[1em] \Rightarrow -\frac{63}{14} \times \frac{a^4b^3c^6}{a^2b^5c^4} \\[1em] \Rightarrow -\frac{9}{2} \times \frac{a^{(4-2)}c^{(6-4)}}{b^{(5-3)}} \\[1em] \Rightarrow -\frac{9a^2c^2}{2b^2}\end{aligned}$$

Hence, the required quotient = $\bm{-\frac{9a^2c^2}{2b^2}}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{6ax - 3cx + 15x}{-3x} \\[1em] \Rightarrow \frac{6ax}{-3x} - \frac{3cx}{-3x} + \frac{15x}{-3x} \\[1em] \Rightarrow -2a + c - 5\end{aligned}$$

Hence, the required quotient = $\bm{-2a + c - 5}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{10mn^2 - 15m^2n^2 + 5m^3n}{-5mn} \\[1em] \Rightarrow \frac{10mn^2}{-5mn} - \frac{15m^2n^2}{-5mn} + \frac{5m^3n}{-5mn} \\[1em] \Rightarrow -2n^{(2-1)} + 3m^{(2-1)}n^{(2-1)} - m^{(3-1)} \\[1em] \Rightarrow -2n + 3mn - m^2\end{aligned}$$

Hence, the required quotient = $\bm{-2n + 3mn - m^2}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{14x^3y^4 - 7x^4y^3 - 28x^3y^6}{-7x^3y^2} \\[1em] \Rightarrow \frac{14x^3y^4}{-7x^3y^2} - \frac{7x^4y^3}{-7x^3y^2} - \frac{28x^3y^6}{-7x^3y^2} \\[1em] \Rightarrow -2y^{(4-2)} + x^{(4-3)}y^{(3-2)} + 4y^{(6-2)} \\[1em] \Rightarrow -2y^2 + xy + 4y^4\end{aligned}$$

Hence, the required quotient = $\bm{-2y^2 + xy + 4y^4}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{18a^6b^3 - 30a^4b^5 + 6a^4b^4}{6a^2b^2} \\[1em] \Rightarrow \frac{18a^6b^3}{6a^2b^2} - \frac{30a^4b^5}{6a^2b^2} + \frac{6a^4b^4}{6a^2b^2} \\[1em] \Rightarrow 3a^{(6-2)}b^{(3-2)} - 5a^{(4-2)}b^{(5-2)} + a^{(4-2)}b^{(4-2)} \\[1em] \Rightarrow 3a^4b - 5a^2b^3 + a^2b^2\end{aligned}$$

Hence, the required quotient = $\bm{3a^4b - 5a^2b^3 + a^2b^2}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{8a^3b - 16a^2b^2 - 6a^4b^3}{-2a^2b} \\[1em] \Rightarrow \frac{8a^3b}{-2a^2b} - \frac{16a^2b^2}{-2a^2b} - \frac{6a^4b^3}{-2a^2b} \\[1em] \Rightarrow -4a^{(3-2)} + 8b^{(2-1)} + 3a^{(4-2)}b^{(3-1)} \\[1em] \Rightarrow -4a + 8b + 3a^2b^2\end{aligned}$$

Hence, the required quotient = $\bm{-4a + 8b + 3a^2b^2}$.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \Big(\frac{1}{2}p^2q^3 - \frac{5}{8}p^3q^2 + \frac{1}{4}p^3q^3\Big) \div \Big(-\frac{1}{4}p^2q^2\Big) \\[1em] \Rightarrow \frac{\frac{1}{2}p^2q^3}{-\frac{1}{4}p^2q^2} - \frac{\frac{5}{8}p^3q^2}{-\frac{1}{4}p^2q^2} + \frac{\frac{1}{4}p^3q^3}{-\frac{1}{4}p^2q^2} \\[1em] \Rightarrow -2q^{(3-2)} + \frac{5}{2}p^{(3-2)} - p^{(3-2)}q^{(3-2)} \\[1em] \Rightarrow -2q + \frac{5}{2}p - pq\end{aligned}$$

Hence, the required quotient = $\bm{-2q + \frac{5}{2}p - pq}$.

Dividing $x^2 + 8x + 15$ by $x + 5$

Hence, quotient = $\bm{x + 3}$ and remainder = 0.

Dividing $4x^2 + 11x - 3$ by $x + 3$

Hence, quotient = $\bm{4x - 1}$ and remainder = 0.

Arranging the terms of the dividend in descending powers of $x$, we divide $-2x^2 + x + 3$ by $x + 1$

Hence, quotient = $\bm{-2x + 3}$ and remainder = 0.

Dividing $x^3 - 9x^2 + 26x - 24$ by $x - 4$

Hence, quotient = $\bm{x^2 - 5x + 6}$ and remainder = 0.

Dividing $x^3 - 3x^2y + 3xy^2 - y^3$ by $x - y$

Hence, quotient = $\bm{x^2 - 2xy + y^2}$ and remainder = 0.

Dividing $12x^2 + 11x + 2$ by $4x + 1$

Hence, quotient = $\bm{3x + 2}$ and remainder = 0.

Dividing $6x^2 + x - 15$ by $2x - 3$

Hence, quotient = $\bm{3x + 5}$ and remainder = 0.

Dividing $6x^3 + x^2 - 26x - 21$ by $3x - 7$

Hence, quotient = $\bm{2x^2 + 5x + 3}$ and remainder = 0.

Dividing $12x^2 + 7xy - 12y^2$ by $3x + 4y$

Hence, quotient = $\bm{4x - 3y}$ and remainder = 0.

Writing the missing power of $x$ in the dividend with coefficient 0, we divide $x^3 + 0x^2 - 2x - 1$ by $x^2 - x - 1$

Hence, quotient = $\bm{x + 1}$ and remainder = 0.

Dividing $x^3 - 6x^2 + 11x - 6$ by $x^2 - 5x + 6$

Hence, quotient = $\bm{x - 1}$ and remainder = 0.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $6x^5 + 4x^4 - 3x^3 + 0x^2 + 0x - 1$ by $3x^2 - x + 1$

Hence, quotient = $\bm{2x^3 + 2x^2 - x - 1}$ and remainder = 0.

Writing the missing power of $x$ in the dividend with coefficient 0, we divide $6x^5 + 0x^4 - 28x^3 + 3x^2 + 30x - 9$ by $2x^2 - 6$

Hence, quotient = $\bm{3x^3 - 5x + \frac{3}{2}}$ and remainder = 0.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $x^3 + 0x^2 + 0x + 27$ by $x + 3$

Hence, quotient = $\bm{x^2 - 3x + 9}$ and remainder = 0.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $x^4 + 0x^3 + 0x^2 + 0x - 81$ by $x + 3$

Hence, quotient = $\bm{x^3 - 3x^2 + 9x - 27}$ and remainder = 0.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $27x^3 + 0x^2 + 0x - 8$ by $3x - 2$

Hence, quotient = $\bm{9x^2 + 6x + 4}$ and remainder = 0.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x - 8$ by $x^2 - 2$

Hence, quotient = $\bm{x^4 + 2x^2 + 4}$ and remainder = 0.

Writing the missing terms of the dividend with coefficient 0, we divide $x^6 + 0x^5y + 0x^4y^2 + 0x^3y^3 + 0x^2y^4 + 0xy^5 - y^6$ by $x - y$

Hence, quotient = $\bm{x^5 + x^4y + x^3y^2 + x^2y^3 + xy^4 + y^5}$ and remainder = 0.

Writing the missing terms of the dividend with coefficient 0, we divide $16x^4 + 0x^3y + 0x^2y^2 + 0xy^3 - 81y^4$ by $2x - 3y$

Hence, quotient = $\bm{8x^3 + 12x^2y + 18xy^2 + 27y^3}$ and remainder = 0.

Dividing $6a^2 - 31a + 47$ by $2a - 5$

Hence, quotient = $\bm{3a - 8}$ and remainder = 7.

Dividing $2x^4 - x^3 + 10x^2 + 8x - 5$ by $x^2 - x + 6$

Hence, quotient = $\bm{2x^2 + x - 1}$ and remainder = $\bm{x + 1}$.

Arranging the terms of the divisor in descending powers of $t$, we divide $3t^5 + 7t^4 - 11t^3 + 8t^2 - 32t + 5$ by $t^2 + 3t + 2$

Hence, quotient = $\bm{3t^3 - 2t^2 - 11t + 45}$ and remainder = $\bm{-145t - 85}$.

Writing the missing powers of $x$ in the dividend with coefficient 0, we divide $x^6 + 0x^5 + 0x^4 + 0x^3 + 3x^2 + 0x + 10$ by $x^3 + 1$

Hence, quotient = $\bm{x^3 - 1}$ and remainder = $\bm{3x^2 + 11}$.

A polynomial is a factor of another polynomial if it divides that polynomial exactly, that is, if the remainder is 0.

Dividing $6a^5 - a^4 + 4a^3 - 5a^2 - a - 15$ by $2a^2 - a + 3$

Here, the remainder is 0.

Hence, $\bm{(2a^2 - a + 3)}$ is a factor of $\bm{(6a^5 - a^4 + 4a^3 - 5a^2 - a - 15)}$.

If a polynomial is divided by another polynomial, then on subtracting the remainder from the dividend, the resulting polynomial becomes exactly divisible by the divisor.

So, we first find the remainder on dividing $8x^4 + 14x^3 - 2x^2 + 7x - 8$ by $4x^2 + 3x - 2$.

Here, quotient = $2x^2 + 2x - 1$ and remainder = $14x - 10$

Hence, $\bm{14x - 10}$ must be subtracted from the given polynomial.

Length of the rectangle = $(a + 5b)$ units

Breadth of the rectangle = $(7a - b)$ units

Perimeter of the rectangle = 2 $\times$ (length + breadth)

$$\begin{aligned}= 2 \times [(a + 5b) + (7a - b)] \\[1em] = 2 \times (8a + 4b) \\[1em] = (16a + 8b) \text{ units}\end{aligned}$$

As the perimeter of the square is equal to the perimeter of the rectangle,

Side of the square = $\frac{\text{Perimeter of the square}}{4}$

$$\begin{aligned}= \frac{16a + 8b}{4} \\[1em] = (4a + 2b) \text{ units}\end{aligned}$$

Now we find the area of the square.

So, area of the square = $(16a^2 + 16ab + 4b^2)$ square units.

Next we find the area of the rectangle.

So, area of the rectangle = $(7a^2 + 34ab - 5b^2)$ square units.

Now we subtract the area of the rectangle from the area of the square.

Hence, the area of the rectangle is $\bm{(9a^2 - 18ab + 9b^2)}$ square units less than the area of the square.

Total sum = ₹ $(16x^3 - 46x^2 + 39x - 9)$

Number of persons = $(8x - 3)$

∴ Amount received by each person = Total sum ÷ Number of persons

Dividing $16x^3 - 46x^2 + 39x - 9$ by $8x - 3$

Hence, the amount received by each person = ₹ $\bm{(2x^2 - 5x + 3)}$.

Product of the two numbers = $(x^6 - y^6)$

One number = $(x - y)$

∴ Other number = Product ÷ One number

Writing the missing terms of the dividend with coefficient 0, we divide $x^6 + 0x^5y + 0x^4y^2 + 0x^3y^3 + 0x^2y^4 + 0xy^5 - y^6$ by $x - y$

Hence, the other number = $\bm{x^5 + x^4y + x^3y^2 + x^2y^3 + xy^4 + y^5}$.

First we find the product of $2x - 3$ and $x^2 - 2x + 2$.

So, the divisor is $2x^3 - 7x^2 + 10x - 6$.

Now we divide $6x^4 - 13x^3 + 2x^2 + 22x - 24$ by $2x^3 - 7x^2 + 10x - 6$

Hence, quotient = $\bm{3x + 4}$ and remainder = 0.

Any factor of a term is called the coefficient of the product of all the remaining factors.

$$\begin{aligned}\Rightarrow \frac{-2}{5}xy^2z^3 \\[1em] \Rightarrow y^2 \times \frac{-2}{5}xz^3\end{aligned}$$

So, the coefficient of $y^2$ is $\frac{-2}{5}xz^3$.

Hence, option 4 is the correct option.

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

I. In $x + \frac{2}{x}$, the term $\frac{2}{x} = 2x^{-1}$, in which the power of $x$ is -1, which is not a non-negative integer. So, it is not a polynomial.

II. In $x^3 - y^3 + xy^2 - x^3y$, the powers of $x$ and $y$ in each term are non-negative integers. So, it is a polynomial.

III. In $\sqrt{2}x^2 + \sqrt{3}y^2$, the powers of $x$ and $y$ are 2 and 2. Here $\sqrt{2}$ and $\sqrt{3}$ are only constant coefficients. So, it is a polynomial.

IV. In $7x + 5\sqrt{y} + 3z^2$, the term $5\sqrt{y}$ has $y$ raised to the power $\frac{1}{2}$, which is not a non-negative integer. So, it is not a polynomial.

V. $\sqrt{x^4} + \sqrt{y^2} + z = x^2 + y + z$, in which the powers of the variables are non-negative integers. So, it is a polynomial.

So, II, III and V are polynomials.

Hence, option 2 is the correct option.

An algebraic expression containing two terms is called a binomial. Multiplication and division do not separate the terms, and like terms must first be added.

Option 1: $2x \times 3y = 6xy$, which has only one term. So, it is a monomial.

Option 2: In $3x^2y + 4xy^2$, the two terms are unlike terms, so they cannot be added. It has two terms and so it is a binomial.

Option 3: $8x^2y + 16yx^2 = (8 + 16)x^2y = 24x^2y$, which has only one term. So, it is a monomial.

Option 4: $-5x^2yz^2 + 5z^2yx^2 = (-5 + 5)x^2yz^2 = 0$, so it is not a binomial.

Hence, option 2 is the correct option.

If a polynomial involves two or more variables, then the sum of the powers of all the variables in each term is taken up and the highest sum so obtained is the degree of the polynomial.

Sum of powers of $a^6 = 6$

Sum of powers of $b^6 = 6$

Sum of powers of $-a^3b^4 = 3 + 4 = 7$

Sum of powers of $a^4b^3 = 4 + 3 = 7$

The greatest sum is 7.

Hence, option 4 is the correct option.

Two or more polynomials can be added by arranging them so that the like terms lie in the same vertical column and adding each column beginning from the left.

Arranging columnwise with like terms in the same column and adding, we get :

$$\begin{array}{rrrrr} & 6a & + 4b & - c & + 3 \\ & - 5a & & + 2c & - 6 \\ & & + 2b & - 3c & + 4 \</p> <ul> <li>& - 7a & + 11b & + 2c & - 1 \\ \hline & - 6a & + 17b & & \\ \end{array}$$

So, the required sum is $(-6a + 17b)$.

Hence, option 2 is the correct option.

Rearrange the terms of the given polynomials with like terms occurring in the same vertical column. Now, change the sign of each term of the polynomial to be subtracted and add the two polynomials.

Arranging columnwise with like terms in the same column, we get :

$$\begin{array}{rrrrrl} & 7p^2 & + 3q & - 2r^3 & + 4 & \\ & 4p^2 & - 2q & + 7r^3 & - 3 & \\ &- &+ &- &+ & \\ \hline & 3p^2 & + 5q & - 9r^3 & + 7 & \\ \end{array}$$

So, the required difference is $(3p^2 + 5q - 9r^3 + 7)$.

Hence, option 1 is the correct option.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging columnwise and multiplying, we get :

$$\begin{array}{rrrrl} & & x & + 5 & \\ \times & & x & - 3 & \\ \hline & x^2 & + 5x & & \quad \text{[Multiplying by } x \text{]} \\ & & - 3x & - 15 & \quad \text{[Multiplying by } -3 \text{]} \\ \hline & x^2 & + 2x & - 15 & \\ \end{array}$$

Hence, option 2 is the correct option.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging columnwise and multiplying, we get :

$$\begin{array}{rrrrl} & & 2x & + 3 & \\ \times & & 3x & - 1 & \\ \hline & 6x^2 & + 9x & & \quad \text{[Multiplying by } 3x \text{]} \\ & & - 2x & - 3 & \quad \text{[Multiplying by } -1 \text{]} \\ \hline & 6x^2 & + 7x & - 3 & \\ \end{array}$$

Hence, option 3 is the correct option.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging columnwise and multiplying, we get :

$$\begin{array}{rrrrl} & & 2x & + 5 & \\ \times & & 2x & - 5 & \\ \hline & 4x^2 & + 10x & & \quad \text{[Multiplying by } 2x \text{]} \\ & & - 10x & - 25 & \quad \text{[Multiplying by } -5 \text{]} \\ \hline & 4x^2 & & - 25 & \\ \end{array}$$

Hence, option 3 is the correct option.

Quotient of two monomials = (Quotient of their numerical coefficients) × (Quotient of their variable parts)

$$\begin{aligned}\Rightarrow \frac{8x^2y^3}{-2xy} \\[1em] \Rightarrow -\frac{8}{2} \times \frac{x^2y^3}{xy} \\[1em] \Rightarrow -4x^{(2-1)}y^{(3-1)} \\[1em] \Rightarrow -4xy^2\end{aligned}$$

Hence, option 2 is the correct option.

Dividing $2x^2 + 3x + 1$ by $x + 1$

$$\begin{array}{l} x + 1\overline{\smash{\big)}2x^2 + 3x + 1}\big(2x + 1 \\ \phantom{x + 1\big)}\underline{2x^2 + 2x \phantom{+ 1}} \\ \phantom{x + 1\big)2x^2;()}x + 1 \\ \phantom{x + 1\big)2x^2;()}\underline{x + 1} \\ \phantom{x + 1\big)2x^2;()}\underset{\displaystyle\hphantom{x + 1}}{0} \end{array}$$

So, the required quotient is $(2x + 1)$.

Hence, option 3 is the correct option.

Dividing $x^2 - 4x + 4$ by $x - 2$

$$\begin{array}{l} x - 2\overline{\smash{\big)}x^2 - 4x + 4}\big(x - 2 \\ \phantom{x - 2\big)}\underline{x^2 - 2x \phantom{+ 4}} \\ \phantom{x - 2\big)x^2;}-2x + 4 \\ \phantom{x - 2\big)x^2;)}\underline{-2x + 4} \\ \phantom{x - 2\big)x^2;)}\underset{\displaystyle\hphantom{-2x + 4}}{0} \end{array}$$

So, the required quotient is $(x - 2)$.

Hence, option 4 is the correct option.

In $2x^3 + 5x^2 - 6$, the term -6 has no literal factor. It can be written as $-6x^0$, in which the power of $x$ is 0. So, the degree of the term -6 is 0.

∴ Assertion (A) is true.

A polynomial having one term consisting of a constant only is called a constant polynomial and the degree of a constant polynomial is 0.

∴ Reason (R) is true.

The term -6 is a constant, and the Reason (R) explains exactly why its degree is 0. So, the Reason (R) is the correct explanation of the Assertion (A).

Hence, option 1 is the correct option.

Multiply each term of the multiplicand by each term of the multiplier and take the algebraic sum of these products.

Arranging columnwise and multiplying, we get :

$$\begin{aligned}\Rightarrow (x + 5) \times (1 - x^2) \\[1em] \Rightarrow x \times 1 - x \times x^2 + 5 \times 1 - 5 \times x^2 \\[1em] \Rightarrow x - x^3 + 5 - 5x^2\end{aligned}$$

Taking -1 as a common factor, this can also be written as $-(x^3 + 5x^2 - x - 5)$.

∴ Assertion (A) is true.

The distributive law for three monomials $a, b$ and $c$ is $(a + b) \times c = ac + bc$, and not $(a \times b) + c = ac + bc$.

∴ Reason (R) is false.

Hence, option 3 is the correct option.

For dividing a polynomial by a monomial, we divide each term of the polynomial by the monomial.

$$\begin{aligned}\Rightarrow \frac{x^3 - 3x^2 + 5x}{x} \\[1em] \Rightarrow \frac{x^3}{x} - \frac{3x^2}{x} + \frac{5x}{x} \\[1em] \Rightarrow x^2 - 3x + 5\end{aligned}$$

∴ Assertion (A) is true.

Multiplication and division are inverse processes of each other.

∴ Reason (R) is true.

But the Assertion (A) is obtained by dividing each term of the polynomial by the monomial, and not by using the inverse relation between multiplication and division. So, the Reason (R) is not the correct explanation of the Assertion (A).

Hence, option 2 is the correct option.