Algebraic Expressions and Identities — Exercise 10.1

Identify monomials, binomials and trinomials from the following algebraic expression:

$5p × q × r^2$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$5p × q × r^2$ is a product of factors, so it has only one term.

Hence, $\bm{5p × q × r^2}$ is a monomial.

Identify monomials, binomials and trinomials from the following algebraic expression:

$3x^2 \times y \div 2z$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$3x^2 \times y \div 2z$ involves only multiplication and division, so it has only one term.

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

Identify monomials, binomials and trinomials from the following algebraic expression:

$-3 + 7x^2$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$-3 + 7x^2$ has two terms, namely -3 and $7x^2$.

Hence, $\bm{-3 + 7x^2}$ is a binomial.

Identify monomials, binomials and trinomials from the following algebraic expression:

$\dfrac{5a^2 - 3b^2 + c}{2}$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$\dfrac{5a^2 - 3b^2 + c}{2} = \dfrac{5a^2}{2} - \dfrac{3b^2}{2} + \dfrac{c}{2}$, which has three terms.

Hence, $\bm{\dfrac{5a^2 - 3b^2 + c}{2}}$ is a trinomial.

Identify monomials, binomials and trinomials from the following algebraic expression:

$7x^5 - \dfrac{3x}{y}$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$7x^5 - \dfrac{3x}{y}$ has two terms, namely $7x^5$ and $-\dfrac{3x}{y}$.

Hence, $\bm{7x^5 - \dfrac{3x}{y}}$ is a binomial.

Identify monomials, binomials and trinomials from the following algebraic expression:

$5p \div 3q - 3p^2 \times q^2$

Answer

Multiplication and division do not separate the terms of an algebraic expression. Only the parts separated by '+' or '-' sign are counted as terms.

$5p \div 3q - 3p^2 \times q^2$ has two terms, namely $5p \div 3q$ and $3p^2 \times q^2$.

Hence, $\bm{5p \div 3q - 3p^2 \times q^2}$ is a binomial.

Identify which of the following expressions are polynomials. If so write their degrees:

(i) $\dfrac{2}{5}x^4 - \sqrt{3}x^2 + 5x - 1$

(ii) $7x^3 - \dfrac{3}{x^2} + \sqrt{5}$

(iii) $4a^3b^2 - 3ab^4 + 5ab + \dfrac{2}{3}$

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

Answer

An algebraic expression is a polynomial if the powers of the variables in each term are non-negative integers. The degree is the greatest sum of the powers of the variables in a term.

(i) $\dfrac{2}{5}x^4 - \sqrt{3}x^2 + 5x - 1$

The powers of $x$ in the terms are 4, 2, 1 and 0, which are all non-negative integers.

So, it is a polynomial and the greatest power is 4.

Hence, it is a polynomial of degree 4.

(ii) $7x^3 - \dfrac{3}{x^2} + \sqrt{5}$

Here $\dfrac{3}{x^2} = 3x^{-2}$, in which the power of $x$ is -2, a negative integer.

Hence, it is not a polynomial.

(iii) $4a^3b^2 - 3ab^4 + 5ab + \dfrac{2}{3}$

The powers of $a$ and $b$ in each term are non-negative integers, so it is a polynomial.

The sums of the powers of the terms are 3 + 2 = 5, 1 + 4 = 5, 1 + 1 = 2 and 0.

The greatest sum is 5.

Hence, it is a polynomial of degree 5.

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

Here $\dfrac{3}{xy} = 3x^{-1}y^{-1}$, in which the powers of $x$ and $y$ are negative integers.

Hence, it is not a polynomial.

Add the following expressions:

$ab - bc, bc - ca, ca - ab$

Answer

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrr} & ab & - bc & \\ & & + bc & - ca \\ + & - ab & & + ca \\ \hline & 0 & & \\ \end{array}$

Hence, the required sum = 0.

Add the following expressions:

$5p^2q^2 + 4pq + 7, 3 + 9pq - 2p^2q^2$

Answer

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrr} & 5p^2q^2 & + 4pq & + 7 \\ + & - 2p^2q^2 & + 9pq & + 3 \\ \hline & 3p^2q^2 & + 13pq & + 10 \\ \end{array}$

Hence, the required sum = $\bm{3p^2q^2 + 13pq + 10}$.

Add the following expressions:

$l^2 + m^2 + n^2, lm + mn, mn + nl, nl + lm$

Answer

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrrrrr} & l^2 & + m^2 & + n^2 & & & \\ & & & & lm & + mn & \\ & & & & & + mn & + nl \\ + & & & & + lm & & + nl \\ \hline & l^2 & + m^2 & + n^2 & + 2lm & + 2mn & + 2nl \\ \end{array}$

The sum can also be written as $l^2 + m^2 + n^2 + 2(lm + mn + nl)$.

Hence, the required sum = $\bm{l^2 + m^2 + n^2 + 2(lm + mn + nl)}$.

Add the following expressions:

$4x^3 - 7x^2 + 9, 3x^2 - 5x + 4, 7x^3 - 11x + 1, 6x^2 - 13x$

Answer

Write each expression in a separate row, keeping the like terms one below the other, and then add them.

$\begin{array}{rrrrr} & 4x^3 & - 7x^2 & & + 9 \\ & & + 3x^2 & - 5x & + 4 \\ & 7x^3 & & - 11x & + 1 \\ + & & + 6x^2 & - 13x & \\ \hline & 11x^3 & + 2x^2 & - 29x & + 14 \\ \end{array}$

Hence, the required sum = $\bm{11x^3 + 2x^2 - 29x + 14}$.

Subtract:

$8a + 3ab - 2b + 7$ from $14a - 5ab + 7b - 5$

Answer

Subtracting,

$\begin{array}{rrrrr} & 14a & - 5ab & + 7b & - 5 \\ & 8a & + 3ab & - 2b & + 7 \\ &- &- &+ &- \\ \hline & 6a & - 8ab & + 9b & - 12 \\ \end{array}$

Hence, the required difference = $\bm{6a - 8ab + 9b - 12}$.

Subtract:

$8xy + 4yz + 5zx$ from $12xy - 3yz - 4zx + 5xyz$

Answer

Subtracting,

$\begin{array}{rrrrr} & 12xy & - 3yz & - 4zx & + 5xyz \\ & 8xy & + 4yz & + 5zx & \\ &- &- &- & \\ \hline & 4xy & - 7yz & - 9zx & + 5xyz \\ \end{array}$

Hence, the required difference = $\bm{4xy - 7yz - 9zx + 5xyz}$.

Subtract:

$4p^2q - 3pq + 5pq^2 - 8p + 7q - 10$ from $18 - 3p - 11q + 5pq - 2pq^2 + 5p^2q$

Answer

Subtracting,

$\begin{array}{rrrrrrr} & 5p^2q & - 2pq^2 & + 5pq & - 3p & - 11q & + 18 \\ & 4p^2q & + 5pq^2 & - 3pq & - 8p & + 7q & - 10 \\ &- &- &+ &+ &- &+ \\ \hline & p^2q & - 7pq^2 & + 8pq & + 5p & - 18q & + 28 \\ \end{array}$

Hence, the required difference = $\bm{p^2q + 8pq - 7pq^2 + 5p - 18q + 28}$.

Subtract the sum of $3x^2 + 5xy + 7y^2 + 3$ and $2x^2 - 4xy - 3y^2 + 7$ from $9x^2 - 8xy + 11y^2$

Answer

First we find the sum of the two given expressions by the column method.

$\begin{array}{rrrrr} & 3x^2 & + 5xy & + 7y^2 & + 3 \\ + & 2x^2 & - 4xy & - 3y^2 & + 7 \\ \hline & 5x^2 & + xy & + 4y^2 & + 10 \\ \end{array}$

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

$\begin{array}{rrrrr} & 9x^2 & - 8xy & + 11y^2 & \\ & 5x^2 & + xy & + 4y^2 & + 10 \\ &- &- &- &- \\ \hline & 4x^2 & - 9xy & + 7y^2 & - 10 \\ \end{array}$

Hence, the required difference = $\bm{4x^2 - 9xy + 7y^2 - 10}$.

What must be subtracted from $3a^2 - 5ab - 2b^2 - 3$ to get $5a^2 - 7ab - 3b^2 + 3a$?

Answer

Let the required expression be A.

According to the question,

⇒ $(3a^2 - 5ab - 2b^2 - 3) - A = 5a^2 - 7ab - 3b^2 + 3a$

⇒ $A = (3a^2 - 5ab - 2b^2 - 3) - (5a^2 - 7ab - 3b^2 + 3a)$

Subtracting,

$\begin{array}{rrrrrr} & 3a^2 & - 5ab & - 2b^2 & & - 3 \\ & 5a^2 & - 7ab & - 3b^2 & + 3a & \\ &- &+ &+ &- & \\ \hline & - 2a^2 & + 2ab & + b^2 & - 3a & - 3 \\ \end{array}$

Hence, $\bm{-2a^2 + 2ab + b^2 - 3a - 3}$ must be subtracted.

The perimeter of a triangle is $7p^2 - 5p + 11$ and two of its sides are $p^2 + 2p - 1$ and $3p^2 - 6p + 3$. Find the third side of the triangle.

Answer

As the perimeter of a triangle is the sum of its three sides,

Third side = Perimeter - Sum of the other two sides

First we find the sum of the two given sides by the column method.

$\begin{array}{rrrr} & p^2 & + 2p & - 1 \\ + & 3p^2 & - 6p & + 3 \\ \hline & 4p^2 & - 4p & + 2 \\ \end{array}$

Now we subtract this sum from the perimeter $7p^2 - 5p + 11$.

$\begin{array}{rrrr} & 7p^2 & - 5p & + 11 \\ & 4p^2 & - 4p & + 2 \\ &- &+ &- \\ \hline & 3p^2 & - p & + 9 \\ \end{array}$

Hence, the third side of the triangle = $\bm{3p^2 - p + 9}$.