NCERT Exemplar Solutions
Class 9 - Mathematics - CHAPTER 2: POLYNOMIALS

Exercise 2.3: Short Answer Questions

Classify the following polynomials as polynomials in one variable, two variables etc.:

(i) \(x^2 + x + 1\)

(ii) \(y^3 - 5y\)

(iii) \(xy + yz + zx\)

(iv) \(x^2 - 2xy + y^2 + 1\)

Determine the degree of each of the following polynomials:

(i) \(2x - 1\)

(ii) \(-10\)

(iii) \(x^3 - 9x + 3x^5\)

(iv) \(y^3(1 - y^4)\)

For the polynomial \(\dfrac{x^3 + 2x + 1}{5} - \dfrac{7}{2}x^2 - x^6\), write:

(i) the degree of the polynomial

(ii) the coefficient of \(x^3\)

(iii) the coefficient of \(x^6\)

(iv) the constant term

Write the coefficient of \(x^2\) in each of the following:

(i) \(\dfrac{\pi}{6}x + x^2 - 1\)

(ii) \(3x - 5\)

(iii) \((x - 1)(3x - 4)\)

(iv) \((2x - 5)(2x^2 - 3x + 1)\)

Classify the following as a constant, linear, quadratic and cubic polynomials:

(i) \(2 - x^2 + x^3\)

(ii) \(3x^3\)

(iii) \(5t - \sqrt{7}\)

(iv) \(4 - 5y^2\)

(v) \(3\)

(vi) \(2 + x\)

(vii) \(y^3 - y\)

(viii) \(1 + x + x^2\)

(ix) \(t^2\)

(x) \(\sqrt{2}x - 1\)

Give an example of a polynomial, which is:

(i) monomial of degree 1

(ii) binomial of degree 20

(iii) trinomial of degree 2

Find the value of the polynomial \(3x^3 - 4x^2 + 7x - 5\), when \(x = 3\) and also when \(x = -3\).

If \(p(x) = x^2 - 4x + 3\), evaluate: \(p(2) - p(-1) + p\left(\dfrac{1}{2}\right)\).

Find \(p(0)\), \(p(1)\), \(p(-2)\) for the following polynomials:

(i) \(p(x) = 10x - 4x^2 - 3\)

(ii) \(p(y) = (y + 2)(y - 2)\)

Verify whether the following are True or False:

(i) \(-3\) is a zero of \(x - 3\)

(ii) \(-\dfrac{1}{3}\) is a zero of \(3x + 1\)

(iii) \(-\dfrac{4}{5}\) is a zero of \(4 - 5y\)

(iv) 0 and 2 are the zeroes of \(t^2 - 2t\)

(v) \(-3\) is a zero of \(y^2 + y - 6\)

Find the zeroes of the polynomial in each of the following:

(i) \(p(x) = x - 4\)

(ii) \(g(x) = 3 - 6x\)

(iii) \(q(x) = 2x - 7\)

(iv) \(h(y) = 2y\)

Find the zeroes of the polynomial: \(p(x) = (x - 2)^2 - (x + 2)^2\).

By actual division, find the quotient and the remainder when the first polynomial is divided by the second polynomial: \(x^4 + 1\); \(x - 1\).

By Remainder Theorem find the remainder, when \(p(x)\) is divided by \(g(x)\), where:

(i) \(p(x) = x^3 - 2x^2 - 4x - 1\), \(g(x) = x + 1\)

(ii) \(p(x) = x^3 - 3x^2 + 4x + 50\), \(g(x) = x - 3\)

(iii) \(p(x) = 4x^3 - 12x^2 + 14x - 3\), \(g(x) = 2x - 1\)

(iv) \(p(x) = x^3 - 6x^2 + 2x - 4\), \(g(x) = 1 - \dfrac{3}{2}x\)

Check whether \(p(x)\) is a multiple of \(g(x)\) or not:

(i) \(p(x) = x^3 - 5x^2 + 4x - 3\), \(g(x) = x - 2\)

(ii) \(p(x) = 2x^3 - 11x^2 - 4x + 5\), \(g(x) = 2x + 1\)

Show that:

(i) \(x + 3\) is a factor of \(69 + 11x - x^2 + x^3\)

(ii) \(2x - 3\) is a factor of \(x + 2x^3 - 9x^2 + 12\)

Determine which of the following polynomials has \(x - 2\) a factor:

(i) \(3x^2 + 6x - 24\)

(ii) \(4x^2 + x - 2\)

Show that \(p - 1\) is a factor of \(p^{10} - 1\) and also of \(p^{11} - 1\).

For what value of \(m\) is \(x^3 - 2mx^2 + 16\) divisible by \(x + 2\)?

If \(x + 2a\) is a factor of \(x^5 - 4a^2x^3 + 2x + 2a + 3\), find \(a\).

Find the value of \(m\) so that \(2x - 1\) be a factor of \(8x^4 + 4x^3 - 16x^2 + 10x + m\).

If \(x + 1\) is a factor of \(ax^3 + x^2 - 2x + 4a - 9\), find the value of \(a\).

Factorise:

(i) \(x^2 + 9x + 18\)

(ii) \(6x^2 + 7x - 3\)

(iii) \(2x^2 - 7x - 15\)

(iv) \(84 - 2r - 2r^2\)

Factorise:

(i) \(2x^3 - 3x^2 - 17x + 30\)

(ii) \(x^3 - 6x^2 + 11x - 6\)

(iii) \(x^3 + x^2 - 4x - 4\)

(iv) \(3x^3 - x^2 - 3x + 1\)

Using suitable identity, evaluate the following:

(i) \(103^3\)

(ii) \(101 \times 102\)

(iii) \(999^2\)

Factorise the following:

(i) \(4x^2 + 20x + 25\)

(ii) \(9y^2 - 66yz + 121z^2\)

(iii) \(\left(2x + \dfrac{1}{3}\right)^2 - \left(x - \dfrac{1}{2}\right)^2\)

Factorise the following:

(i) \(9x^2 - 12x + 3\)

(ii) \(9x^2 - 12x + 4\)

Expand the following:

(i) \((4a - b + 2c)^2\)

(ii) \((3a - 5b - c)^2\)

(iii) \((-x + 2y - 3z)^2\)

Factorise the following:

(i) \(9x^2 + 4y^2 + 16z^2 + 12xy - 16yz - 24xz\)

(ii) \(25x^2 + 16y^2 + 4z^2 - 40xy + 16yz - 20xz\)

(iii) \(16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz\)

If \(a + b + c = 9\) and \(ab + bc + ca = 26\), find \(a^2 + b^2 + c^2\).

Expand the following:

(i) \((3a - 2b)^3\)

(ii) \(\left(\dfrac{1}{x} + \dfrac{y}{3}\right)^3\)

(iii) \(\left(4 - \dfrac{1}{3x}\right)^3\)

Factorise the following:

(i) \(1 - 64a^3 - 12a + 48a^2\)

(ii) \(8p^3 + \dfrac{12}{5}p^2 + \dfrac{6}{25}p + \dfrac{1}{125}\)

Find the following products:

(i) \(\left(\dfrac{x}{2} + 2y\right)\left(\dfrac{x^2}{4} - xy + 4y^2\right)\)

(ii) \((x^2 - 1)(x^4 + x^2 + 1)\)

Factorise:

(i) \(1 + 64x^3\)

(ii) \(a^3 - 2\sqrt{2}b^3\)

Find the following product: \((2x - y + 3z)(4x^2 + y^2 + 9z^2 + 2xy + 3yz - 6xz)\).

Factorise:

(i) \(a^3 - 8b^3 - 64c^3 - 24abc\)

(ii) \(2\sqrt{2}a^3 + 8b^3 - 27c^3 + 18\sqrt{2}abc\)

Without actually calculating the cubes, find the value of:

(i) \(\left(\dfrac{1}{2}\right)^3 + \left(\dfrac{1}{3}\right)^3 - \left(\dfrac{5}{6}\right)^3\)

(ii) \((0.2)^3 - (0.3)^3 + (0.1)^3\)

Without finding the cubes, factorise:

\((x - 2y)^3 + (2y - 3z)^3 + (3z - x)^3\)

Find the value of:

(i) \(x^3 + y^3 - 12xy + 64\), when \(x + y = -4\)

(ii) \(x^3 - 8y^3 - 36xy - 216\), when \(x = 2y + 6\)

Give possible expressions for the length and breadth of the rectangle whose area is given by \(4a^2 + 4a - 3\).