Exercise 2A RS Aggarwal Class 9 Polynomials Best Solutions

Exercise 2A RS Aggarwal Class 9 contains a total of six questions. The questions are based on the following topics.

Polynomials in one variable
Standard form of a polynomial
Terms of an Algebraic Expression
Polynomials of various degrees: linear, quadratic, cubic, bi-quadratic, etc.
Number of terms in a polynomial: monomial, binomial, trinomial, etc.
Constant polynomial
Zero polynomial

Exercise 2A RS Aggarwal Class 9 Solutions

The first question of Exercise 2A RS Aggarwal Class 9 asks to identify whether a given expression is a polynomial or not.

1. Which of the following expressions are polynomials? In case of a polynomial, write its degree..

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

Answer

The given expression is
$x^{5} - 2 x^{3} + x + \sqrt{3}$

∵ The exponents of the variable $x$ in each term are 5, 3 and 1, all whole numbers.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 5, the degree of the polynomial is 5.

(ii) $y^{3} + \sqrt{3} y$

Answer

The given expression is
$y^{3} + \sqrt{3} y$

∵ The exponents of the variable $y$ in each term are 3 and 1, all whole numbers.
∴ The given expression is a polynomial.

Since the highest power of the variable $y$ is 3, the degree of the polynomial is 3.

(iii) $t^{2} - \frac{2}{5} t + \sqrt{5}$

Answer

The given expression is
$t^{2} - \frac{2}{5} t + \sqrt{5}$

∵ The exponents of the variable $t$ in each term are 2 and 1, both whole numbers.
∴ The given expression is a polynomial.

Since the highest power of the variable $t$ is 2, the degree of the polynomial is 2.

(iv) $x^{100} - 1$

Answer

The given expression is
$x^{100} - 1$

∵ The exponent of the variable $x$ is 100, a whole number.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 100, the degree of the polynomial is 100.

(v) $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$

Answer

The given expression is
$\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$ .

∵ The exponents of the variable $x$ in each term are 2 and 1, both whole numbers.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 2, the degree of the polynomial is 2.

(vi) $x^{- 2} + 2 x^{- 1} + 3$

Answer

The given expression is
$x^{- 2} + 2 x^{- 1} + 3$ .

∵ The exponents of the variable $x$ in each term are −2 and −1, both are not whole numbers.
∴ The given expression is not a polynomial.

(vii) 1

Answer

The given expression is
$1 = 1 x^{0}$ .

∵ The exponent of the variable $x$ is 0, a whole number.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 0, the degree of the polynomial is 0.
This polynomial is also called constant polynomial.

(viii) $\frac{- 3}{5}$

Answer

The given expression is
$\frac{- 3}{5} = \frac{- 3}{5} x^{0}$ .

∵ The exponent of the variable $x$ is 0, a whole number.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 0, the degree of the polynomial is 0.
This polynomial is also called constant polynomial.

(ix) $\frac{x^{2}}{2} - \frac{2}{x^{2}}$

Answer

The given expression is
$\frac{x^{2}}{2} - \frac{2}{x^{2}}$ .

The given expression can be rewritten as
$\frac{x^{2}}{2} - \frac{2}{x^{2}} = \frac{1}{2} x^{2} - 2 x^{- 2}$

∵ One of the exponents of the variable $x$ is −2, it is not a whole number.
∴ The given expression is not a polynomial.

(x) $\sqrt[3]{2} x^{2} - 8$

Answer

The given expression is
$\sqrt[3]{2} x^{2} - 8$ .

∵ The exponent of the variable $x$ is 2, a whole number.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 2, the degree of the polynomial is 2.

(xi) $\frac{1}{2 x^{2}}$

Answer

The given expression is
$\frac{1}{2 x^{2}}$ .

The given expression can be rewritten as
$\frac{1}{2 x^{2}} = \frac{1}{2} x^{- 2}$ .

∵ One of the exponents of the variable $x$ is −2, it is not a whole number.
∴ The given expression is not a polynomial.

(xii) $\frac{1}{\sqrt{5}} x^{1 / 2} + 1$

Answer

The given expression is
$\frac{1}{\sqrt{5}} x^{1 / 2} + 1$

∵ One of the exponents of the variable $x$ is $\frac{1}{2}$ , it is not a whole number.
∴ The given expression is not a polynomial.

(xiii) $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$

Answer

The given expression is
$\frac{3}{5} x^{2} - \frac{7}{3} x + 9$ .

∵ The exponents of the variable $x$ in each term are 2 and 1, both whole numbers.
∴ The given expression is a polynomial.

Since the highest power of the variable $x$ is 2, the degree of the polynomial is 2.

(xiv) $x^{4} - x^{3 / 2} + x - 3$

Answer

The given expression is
$x^{4} - x^{3 / 2} + x - 3$ .

∵ One of the exponents of the variable $x$ is $\frac{3}{2}$ , it is not a whole number.
∴ The given expression is not a polynomial.

(xv) $2 x^{3} + 3 x^{2} + \sqrt{x} - 1$

Answer

The given expression is
$2 x^{3} + 3 x^{2} + \sqrt{x} - 1$ .

The given expression can be rewritten as
$2 x^{3} + 3 x^{2} + \sqrt{x} - 1 = 2 x^{3} + 3 x^{2} + x^{\frac{1}{2}} - 1$

∵ One of the exponents of the variable $x$ is $\frac{1}{2}$ , it is not a whole number.
∴ The given expression is not a polynomial.

The second question of Exercise 2A RS Aggarwal Class 9 asks us to identify constant, linear, quadratic, cubic and quartic polynomials out of the given polynomials.

2. Identify constant, linear, quadratic, cubic and quartic polynomials from the following.

(i) $- 7 + x$

Answer

The given polynomial is
$- 7 + x$ .

Since the degree of the polynomial is 1, it is a linear polynomial.

(ii) $6 y$

Answer

The given polynomial is
$6 y$ .

Since the degree of the polynomial is 1, it is a linear polynomial.

(iii) $- z^{3}$

Answer

The given polynomial is
$- z^{3}$ .

Since the degree of the polynomial is 3, it is a cubic polynomial.

(iv) $1 - y - y^{3}$

Answer

The given polynomial is
$1 - y - y^{3}$ .

Since the degree of the polynomial is 3, it is a cubic polynomial.

(v) $x - x^{3} + x^{4}$

Answer

The given polynomial is
$x - x^{3} + x^{4}$ .

Since the degree of the polynomial is 4, it is a quartic polynomial.

(vi) $1 + x + x^{2}$

Answer

The given polynomial is
$1 + x + x^{2}$ .

Since the degree of the polynomial is 2, it is a quadratic polynomial.

(vii) $- 6 x^{2}$

Answer

The given polynomial is
$- 6 x^{2}$

Since the degree of the polynomial is 2, it is a quadratic polynomial.

(viii) $- 13$

Answer

The given polynomial is
$- 13$

Since the degree of the polynomial is 0, it is a constant polynomial.

(ix) $- p$

Answer

The given polynomial is
$- p$

Since the degree of the polynomial is 1, it is a linear polynomial.

In Exercise 2A RS Aggarwal Class 9, the third question asks us to figure out the value of the coefficients and constant terms in the given polynomial.

3. Write

(i) the coefficient of $x^{3}$ in $x + 3 x^{2} - 5 x^{3} + x^{4}$ .

Answer

The given polynomial is
$x + 3 x^{2} - 5 x^{3} + x^{4}$ .

The coefficient of $x^{3}$ is −5.

(ii) the coefficient of $x$ in $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ .

Answer

The given polynomial is
$\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ .

The coefficient of $x$ is $- 2 \sqrt{2}$ .

(iii) the coefficient of $x^{2}$ in $2 x - 3 + x^{3}$ .

Answer

The given polynomial is
$2 x - 3 + x^{3}$ .

There is no term containing $x^{2}$ in the given polynomial.
Rewriting the given polynomial, we get.

$2 x - 3 + x^{3} = x^{3} + 0 x^{2} + 2 x - 3$

Clearly, the coefficient of $x^{2}$ is 0.

(iv) the coefficient of $x$ in $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ .

Answer

The given polynomial is
$\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ .

The coefficient of $x$ is $- \frac{2}{7}$ .

(v) the constant term in $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ .

Answer

The given polynomial is
$\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ .

The constant term of the polynomial is $- \frac{2}{5} π$ .

The fourth question of exercise 2A RS Aggarwal Class 9 asks us to determine the degree of each of the given polynomials.

4. Determine the degree of each of the following polynomials.

(i) $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x}$ .

Answer

The given polynomial is
$\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x}$ .

Simplifying by distributing the denominator, we get.

$= \frac{4 x}{2 x} - \frac{5 x^{2}}{2 x} + \frac{6 x^{3}}{2 x}$ .

$= 2 - \frac{5}{2} x + 3 x^{2}$ .

Since the highest power of the variable $x$ in the polynomial is 2, its degree is 2.

(ii) $y^{2} (y - y^{3})$ .

Answer

The given polynomial is
$y^{2} (y - y^{3})$ .

$= y^{3} - y^{5}$ .

Since the highest power of the variable $y$ is 5, its degree is 5.

(iii) $(3 x - 2) (2 x^{3} + 3 x^{2})$ .

Answer

The given polynomial is
$(3 x - 2) (2 x^{3} + 3 x^{2})$ .

$= 3 x (2 x^{3} + 3 x^{2}) - 2 (2 x^{3} + 3 x^{2})$ .

$= 6 x^{4} + 9 x^{3} - 4 x^{3} - 6 x^{2}$ .

$= 6 x^{4} + 5 x^{3} - 6 x^{2}$ .

Since the highest power of the polynomial is 4, its degree is 4.

(iv) $- \frac{1}{2} x + 3$

Answer

The given polynomial is
$- \frac{1}{2} x + 3$ .

Since the highest power of the variable $x$ in the polynomial is 1, its degree is 1.

(v) $- 8$

Answer

The given polynomial is
$- 8$ .
= $- 8 x^{0}$ .

(vi) $x^{- 2} (x^{4} + x^{2})$ .

Answer

The given polynomial is
$x^{- 2} (x^{4} + x^{2})$ .

$= x^{- 2} \times x^{4} + x^{- 2} \times x^{2}$

$= x^{- 2 + 4} + x^{- 2 + 2}$

$= x^{2} + x^{0}$

$= x^{2} + 1$

Since the highest power of the variable $x$ is 2, its degree is 2.

The fifth question of Exercise 2A RS Aggarwal Class 9 asks us to give an example of a monomial, a binomial and a trinomial.

5.

(i) Give an example of a monomial of degree 5.

Answer

Monomial is a polynomial having only one term. The examples of monomials of degree 5 are given below.

$2 x^{5}$
$6 x^{5}$
$x^{5}$
$- 3 x^{5}$

(ii) Give an example of a binomial of degree 8.

Answer

A binomial is a polynomial having two terms. The examples of binomials of degree 8 are given below.

$x + x^{8}$
$x^{8} - 2 x^{7}$
$4 x^{8} - \sqrt{3} x^{7}$
$- 7 x^{8} - 3 x^{5}$

(iii) Give an example of a trinomial of degree 4.

Answer

A trinomial is a polynomial having three terms. The examples of trinomials of degree 4 are given below.

$x^{4} + x^{3} + x$
$3 x^{4} - 2 x^{3} + x$
$- 3 x^{4} - \sqrt{5} x^{3} + x^{2}$
$786 x^{4} - \frac{3}{4} x^{3} + 108 x^{2}$

(iv) Give an example of a monomial of degree 0.

Answer

The examples of monomials of degree 0 are given below.

$5$
$1008$
$786$
$- 2$

The sixth question of exercise 2A RS Aggarwal Class 9 asks us to write the given polynomials in standard form.

6. Rewrite each of the following polynomials in standard form.

(i) $x - 2 x^{2} + 8 + 5 x^{3}$

Answer

Writing a polynomial in standard form entails arranging the terms in descending order of their exponents.

The polynomial in standard form is as follows:
$x - 2 x^{2} + 8 + 5 x^{3}$ $=$ $5 x^{3} - 2 x^{2} + x + 8$

(ii) $\frac{2}{3} + 4 y^{2} - 3 y + 2 y^{3}$

Answer

The standard form of polynomial
$\frac{2}{3} + 4 y^{2} - 3 y + 2 y^{3}$

= $2 y^{3} + 4 y^{2} - 3 y + \frac{2}{3}$ .

(iii) $6 x^{3} + 2 x - x^{5} - 3 x^{2}$

Answer

The standard form of polynomial
$6 x^{3} + 2 x - x^{5} - 3 x^{2}$

= $- x^{5} + 6 x^{3} - 3 x^{2} + 2 x$

(iv) $2 + t - 3 t^{3} + t^{4} - t^{2}$

Answer

The standard form of polynomial
$2 + t - 3 t^{3} + t^{4} - t^{2}$

= $t^{4} - 3 t^{3} - t^{2} + t + 2$

Exercise 2A RS Aggarwal Class 9 Solutions

1. Which of the following expressions are polynomials? In case of a polynomial, write its degree..

(i) x5−2x3+x+3

Answer

(ii) y3+3y

Answer

(iii) t2−25t+5

Answer

(iv) x100−1

Answer

(v) 12x2−2x+2

Answer

(vi) x−2+2x−1+3

Answer

(vii) 1

Answer

(viii) −35

Answer

(ix) x22−2x2

Answer

(x) 23x2−8

Answer

(xi) 12x2

Answer

(xii) 15x1/2+1

Answer

(xiii) 35x2−73x+9

Answer

(xiv) x4−x3/2+x−3

Answer

(xv) 2x3+3x2+x−1

Answer

2. Identify constant, linear, quadratic, cubic and quartic polynomials from the following.

(i) −7+x

Answer

(ii) 6y

Answer

(iii) −z3

Answer

(iv) 1−y−y3

Answer

(v) x−x3+x4

Answer

(vi) 1+x+x2

Answer

(vii) −6x2

Answer

(viii) −13

Answer

(ix) −p

Answer

3. Write

(i) the coefficient of x3 in x+3x2−5x3+x4.

Answer

(ii) the coefficient of x in 3−22x+6x2.

Answer

(iii) the coefficient of x2 in 2x−3+x3.

Answer

(iv) the coefficient of x in 38x2-27x+16.

Answer

(v) the constant term in π2x2+7x−25π.

Answer

4. Determine the degree of each of the following polynomials.

(i) 4x−5x2+6x32x.

Answer

(ii) y2(y−y3).

Answer

(iii) (3x−2)(2x3+3x2).

Answer

(iv) −12x+3

Answer

(v) −8

Answer

(vi) x−2(x4+x2).

Answer

5.

(i) Give an example of a monomial of degree 5.

Answer

(ii) Give an example of a binomial of degree 8.

Answer

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

(ii) $y^{3} + \sqrt{3} y$

(iii) $t^{2} - \frac{2}{5} t + \sqrt{5}$

(iv) $x^{100} - 1$

(v) $\frac{1}{\sqrt{2}} x^{2} - \sqrt{2} x + 2$

(vi) $x^{- 2} + 2 x^{- 1} + 3$

(viii) $\frac{- 3}{5}$

(ix) $\frac{x^{2}}{2} - \frac{2}{x^{2}}$

(x) $\sqrt[3]{2} x^{2} - 8$

(xi) $\frac{1}{2 x^{2}}$

(xii) $\frac{1}{\sqrt{5}} x^{1 / 2} + 1$

(xiii) $\frac{3}{5} x^{2} - \frac{7}{3} x + 9$

(xiv) $x^{4} - x^{3 / 2} + x - 3$

(xv) $2 x^{3} + 3 x^{2} + \sqrt{x} - 1$

(i) $- 7 + x$

(ii) $6 y$

(iii) $- z^{3}$

(iv) $1 - y - y^{3}$

(v) $x - x^{3} + x^{4}$

(vi) $1 + x + x^{2}$

(vii) $- 6 x^{2}$

(viii) $- 13$

(ix) $- p$

(i) the coefficient of $x^{3}$ in $x + 3 x^{2} - 5 x^{3} + x^{4}$ .

(ii) the coefficient of $x$ in $\sqrt{3} - 2 \sqrt{2} x + 6 x^{2}$ .

(iii) the coefficient of $x^{2}$ in $2 x - 3 + x^{3}$ .

(iv) the coefficient of $x$ in $\frac{3}{8} x^{2} - \frac{2}{7} x + \frac{1}{6}$ .

(v) the constant term in $\frac{π}{2} x^{2} + 7 x - \frac{2}{5} π$ .

(i) $\frac{4 x - 5 x^{2} + 6 x^{3}}{2 x}$ .

(ii) $y^{2} (y - y^{3})$ .

(iii) $(3 x - 2) (2 x^{3} + 3 x^{2})$ .

(iv) $- \frac{1}{2} x + 3$

(v) $- 8$

(vi) $x^{- 2} (x^{4} + x^{2})$ .

(i) $x - 2 x^{2} + 8 + 5 x^{3}$

(ii) $\frac{2}{3} + 4 y^{2} - 3 y + 2 y^{3}$

(iii) $6 x^{3} + 2 x - x^{5} - 3 x^{2}$

(iv) $2 + t - 3 t^{3} + t^{4} - t^{2}$