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Chapter 13

Fundamental Concepts of Algebra — Exercise 13(A)

Class - 7 Concise Mathematics Selina

Exercise 13(A)

Question 1

Separate constant terms and variable terms from the following:

$8, x, 6xy, 6 + x, -5xy^2, 15az^2, \dfrac{32z}{xy}, \dfrac{y^2}{3x}$

Answer

A term made up of numbers (digits) only is a constant term, whereas a term containing one or more letters (variables) is a variable term.

Constant term = 8

Variable terms = $\bm{x, 6xy, 6 + x, -5xy^2, 15az^2, \dfrac{32z}{xy}}$ and $\bm{\dfrac{y^2}{3x}}$

Question 2

For each expression, given below, state whether it is a monomial, binomial or trinomial:

(i) $2x \div 15$

(ii) $ax + 9$

(iii) $3x^2 \times 5x$

(iv) $5 + 2a - 3b$

(v) $2y - \dfrac{7}{3}z \div x$

(vi) $3p \times q \div z$

(vii) $12z \div 5x + 4$

(viii) $12 - 5z - 4$

(ix) $a^3 - 3ab^2 \times c$

Answer

An expression with 1 term is a monomial, with 2 unlike terms is a binomial and with 3 unlike terms is a trinomial.

(i) $2x \div 15 = \dfrac{2x}{15}$ , which has 1 term → Monomial.

(ii) $ax + 9$ has 2 terms → Binomial.

(iii) $3x^2 \times 5x = 15x^3$ , which has 1 term → Monomial.

(iv) $5 + 2a - 3b$ has 3 terms → Trinomial.

(v) $2y - \dfrac{7}{3}z \div x = 2y - \dfrac{7z}{3x}$ , which has 2 terms → Binomial.

(vi) $3p \times q \div z = \dfrac{3pq}{z}$ , which has 1 term → Monomial.

(vii) $12z \div 5x + 4 = \dfrac{12z}{5x} + 4$ , which has 2 terms → Binomial.

(viii) $12 - 5z - 4 = 8 - 5z$ , which has 2 terms → Binomial.

(ix) $a^3 - 3ab^2 \times c = a^3 - 3ab^2c$ , which has 2 terms → Binomial.

Question 3(i)

Write the coefficient of:

$xy$ in $-3axy$

Answer

$-3axy = (-3a) \times xy$ , so the coefficient of $xy$ is $\bm{-3a}$ .

Question 3(ii)

Write the coefficient of:

$z^2$ in $p^2yz^2$

Answer

$p^2yz^2 = (p^2y) \times z^2$ , so the coefficient of $z^2$ is $\bm{p^2y}$ .

Question 3(iii)

Write the coefficient of:

$mn$ in $-mn$

Answer

$-mn = (-1) \times mn$ , so the coefficient of $mn$ is $\bm{-1}$ .

Question 3(iv)

Write the coefficient of:

15 in $-15p^2$

Answer

$-15p^2 = 15 \times (-p^2)$ , so the coefficient of 15 is $\bm{-p^2}$ .

Question 4(i)

For the following monomials, write its degree:

$7y$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $7y = \bm{1}$ .

Question 4(ii)

For the following monomials, write its degree:

$-x^2y$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $-x^2y = 2 + 1 = \bm{3}$ .

Question 4(iii)

For the following monomials, write its degree:

$xy^2z$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $xy^2z = 1 + 2 + 1 = \bm{4}$ .

Question 4(iv)

For the following monomials, write its degree:

$-9y^2z^3$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $-9y^2z^3 = 2 + 3 = \bm{5}$ .

Question 4(v)

For the following monomials, write its degree:

$3m^3n^4$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $3m^3n^4 = 3 + 4 = \bm{7}$ .

Question 4(vi)

For the following monomials, write its degree:

$-2p^2q^3r^4$

Answer

The degree of a monomial is the sum of the exponents of its variables.

Degree of $-2p^2q^3r^4 = 2 + 3 + 4 = \bm{9}$ .

Question 5(i)

Write the degree of the following polynomial:

$3y^3 - x^2y^2 + 4x$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are 3, (2 + 2) = 4 and 1. The greatest is 4.

Hence, the degree of the polynomial = 4.

Question 5(ii)

Write the degree of the following polynomial:

$p^3q^2 - 6p^2q^5 + p^4q^4$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are (3 + 2) = 5, (2 + 5) = 7 and (4 + 4) = 8. The greatest is 8.

Hence, the degree of the polynomial = 8.

Question 5(iii)

Write the degree of the following polynomial:

$-8mn^6 + 5m^3n$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are (1 + 6) = 7 and (3 + 1) = 4. The greatest is 7.

Hence, the degree of the polynomial = 7.

Question 5(iv)

Write the degree of the following polynomial:

$7 - 3x^2y + y^2$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are 0, (2 + 1) = 3 and 2. The greatest is 3.

Hence, the degree of the polynomial = 3.

Question 5(v)

Write the degree of the following polynomial:

$3x - 15$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are 1 and 0. The greatest is 1.

Hence, the degree of the polynomial = 1.

Question 5(vi)

Write the degree of the following polynomial:

$2y^2z + 9yz^3$

Answer

The degree of a polynomial is the degree of its highest degree term.

Degrees of the terms are (2 + 1) = 3 and (1 + 3) = 4. The greatest is 4.

Hence, the degree of the polynomial = 4.

Question 6(i)

Group the like terms together:

$9x^2, xy, -3x^2, x^2$ and $-2xy$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{9x^2, -3x^2}$ and $\bm{x^2}$ ; $\bm{xy}$ and $\bm{-2xy}$ .

Question 6(ii)

Group the like terms together:

$ab, -a^2b, -3ab, 5a^2b$ and $-8a^2b$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{ab}$ and $\bm{-3ab}$ ; $\bm{-a^2b, 5a^2b}$ and $\bm{-8a^2b}$ .

Question 6(iii)

Group the like terms together:

$7p, 8pq, -5pq, -2p$ and $3p$

Answer

Terms having the same literal (variable) part are like terms.

The groups of like terms are $\bm{7p, -2p}$ and $\bm{3p}$ ; $\bm{8pq}$ and $\bm{-5pq}$ .

Question 7

Write the numerical coefficient of each of the following:

(i) $y$

(ii) $-y$

(iii) $2x^2y$

(iv) $- 8xy^3$

(v) $3py^2$

(vi) $-9a^2b^3$

Answer

(i) $y = 1 \times y$ , so its numerical coefficient is $\bm{1}$ .

(ii) $-y = (-1) \times y$ , so its numerical coefficient is $\bm{-1}$ .

(iii) The numerical coefficient of $2x^2y$ is $\bm{2}$ .

(iv) The numerical coefficient of $-8xy^3$ is $\bm{-8}$ .

(v) The numerical coefficient of $3py^2$ is $\bm{3}$ .

(vi) The numerical coefficient of $-9a^2b^3$ is $\bm{-9}$ .

Question 8

In $-5x^3y^2z^4$ ; write the coefficient of:

(i) $z^2$

(ii) $y^2$

(iii) $yz^2$

(iv) $x^3y$

(v) $-xy^2$

(vi) $-5xy^2z$

Also, write the degree of the given algebraic expression.

Answer

We have $-5x^3y^2z^4$ .

(i) $-5x^3y^2z^4 = (-5x^3y^2z^2) \times z^2$ , so the coefficient of $z^2$ is $\bm{-5x^3y^2z^2}$ .

(ii) $-5x^3y^2z^4 = (-5x^3z^4) \times y^2$ , so the coefficient of $y^2$ is $\bm{-5x^3z^4}$ .

(iii) $-5x^3y^2z^4 = (-5x^3yz^2) \times yz^2$ , so the coefficient of $yz^2$ is $\bm{-5x^3yz^2}$ .

(iv) $-5x^3y^2z^4 = (x^3y)(-5yz^4)$ , so the coefficient of $x^3y$ is $\bm{-5yz^4}$ .

(v) $-5x^3y^2z^4 = (-xy^2)(5x^2z^4)$ , so the coefficient of $-xy^2$ is $\bm{5x^2z^4}$ .

(vi) $-5x^3y^2z^4 = (-5xy^2z)(x^2z^3)$ , so the coefficient of $-5xy^2z$ is $\bm{x^2z^3}$ .

Degree of $-5x^3y^2z^4 = 3 + 2 + 4 = 9$ .

Hence, the degree of the given algebraic expression = 9.

Question 9

Which of the following expressions is a polynomial:

(i) $\dfrac{8}{5}x^3 + 7$

(ii) $8x^4 - 2x^2 + 3\sqrt{x} - 4$

(iii) $8x^3 - \dfrac{8}{x^3} + 15$

(iv) $5x^2 + 3^x - 4x + 18$

(v) $7x^2y^3 - \dfrac{18}{x^3} + \dfrac{7}{y^2} + 4xy$

Answer

An algebraic expression is a polynomial only if the exponent of each variable in every term is a whole number.

(i) In $\dfrac{8}{5}x^3 + 7$ , the exponent of $x$ is 3 which is a whole number → It is a polynomial.

(ii) $3\sqrt{x} = 3x^{\frac{1}{2}}$ has a fractional exponent → Not a polynomial.

(iii) $\dfrac{8}{x^3} = 8x^{-3}$ has a negative exponent → Not a polynomial.

(iv) In $3^x$ , the exponent itself is a variable → Not a polynomial.

(v) $\dfrac{18}{x^3} = 18x^{-3} \text{ and } \dfrac{7}{y^2} = 7y^{-2}$ have negative exponents → Not a polynomial.

Hence, only expression (i) is a polynomial.

Question 10(i)

Generate algebraic expression for:

15 added to the sum of $x$ and $y$

Answer

Sum of $x$ and $y = x + y$ .

Adding 15 to it, the required expression is $\bm{(x + y) + 15}$ .

Question 10(ii)

Generate algebraic expression for:

23 subtracted from the sum of $2m$ and $3y$

Answer

Sum of $2m$ and $3y = 2m + 3y$ .

Subtracting 23 from it, the required expression is $\bm{(2m + 3y) - 23}$ .

Question 10(iii)

Generate algebraic expression for:

product of $a$ and $b$ added to their sum

Answer

Sum of $a$ and $b = a + b$ and their product = $ab$ .

So, the required expression is $\bm{(a + b) + ab}$ .

Question 10(iv)

Generate algebraic expression for:

sum of $x$ and $y$ is subtracted from their product

Answer

Product of $x$ and $y = xy$ and their sum = $x + y$ .

So, the required expression is $\bm{xy - (x + y)}$ .

Question 10(v)

Generate algebraic expression for:

addition of $x \div y$ and $y \div x$ .

Answer

$x \div y = \dfrac{x}{y}$ and $y \div x = \dfrac{y}{x}$

So, the required expression is $\bm{\dfrac{x}{y} + \dfrac{y}{x} = (x \div y) + (y \div x)}$ .