R s aggarwal solutions mathematics class 9 Solutions for Chapter 2 polynomials

Exercises in Chapter 2 polynomials Grade 9

Polynomials Exercise 2A

Polynomials Exercise 2B

Polynomials Exercise 2C

Polynomials Exercise 2D

Questions in Polynomials Exercise 2A

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

(i) – 7 + x

(ii) 6y

(iii) - 𝒛^𝟑

(iv) 1 – y - 𝒚^𝟑

(v) x - 𝒙^𝟑 + 𝒙^𝟒

(vi) 1 + x + 𝒙^𝟐

(vii) -6𝒙^𝟐

(viii) -13

(ix) –p

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

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

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

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

Rewrite each of the following polynomials in standard form.

(i) x – 2𝐱^𝟐 + 8 + 5𝐱^𝟑

(ii) 𝟐/𝟑 + 4𝐲^𝟐- 3y + 2𝐲^𝟑

(iii) 𝟔𝐱^𝟑 + 2x - 𝐱^𝟓- 3𝐱^𝟐

(iv) 𝟐 + 𝒕 − 𝟑𝐭^𝟑+ 𝐭^𝟒 - 𝐭^𝟐

Questions in Polynomials Exercise 2B

If p(x) = 5 – 4x + 2𝒙^𝟐, find

(i) p(0)

(ii) p(3)

(iii) p(-2)

If p(y) = 4 + 3y - 𝒚^𝟐+ 5𝒚^𝟑, find

(i) p(0)

(ii) p(2)

(iii) p(-1)

If f(t) = 4𝒕^𝟐 – 3t + 6, find

(i) f(0)

(ii) f(4)

(iii) f(-5)

If p(x) = 𝒙^𝟑- 3𝒙^𝟐+ 2x, find p (0), p (1), p (2). What do you conclude?

If p(x) = 𝒙^𝟑+ 𝒙^𝟐- 9x – 9, find p (0), p (3), p (-3) and p (-1). What do you conclude about the zeros of p(x)? Is 0 a zero of p(x)?

Verify that

(i) 4 is a zero of the polynomial, p(x) = x – 4.

(ii) -3 is a zero of the polynomial, q(x) = x + 3.

(iii)𝟐/𝟓 is a zero of the polynomial, f(x) = 2-5x.

(iv)−𝟏/𝟐 is a zero of the polynomial, g(y) = 2y+1.

Verify that

(i) 1 and 2 are the zeros of the polynomial, p(x) = 𝒙^𝟐- 3x + 2.

(ii) 2 and -3 are the zeros of the polynomial, q(x) = 𝒙^𝟐+x -6.

(iii) 0 and 3 are the zeros of the polynomial, r(x) = 𝒙^𝟐- 3x.

Find the zero of the polynomial:

(i) p(x) = x-5

(ii) q(x) = x+4

(iii) r(x) = 2x+5

(iv) f(x) = 3x+1

(v) g(x) = 5 - 4x

(vi) h(x) = 6x – 2

(vii) p(x) = ax, a ≠0

(viii) q(x) = 4x

If 2 and 0 are the zeros of the polynomial f(x) = 2𝒙^𝟑- 5𝒙^𝟐+ ax + b then find the values of ‘a’ and ‘b’.

Questions in Polynomials Exercise 2C

By actual division, find the quotient and the remainder when (𝒙^𝟒 + 𝟏) is divided by (x-1). Verify that remainder = f(1).

Verify the division algorithm for the polynomials

p(x) = 2𝒙^𝟒 − 𝟔𝒙^𝟑 + 𝟐𝒙^𝟐 − 𝒙 + 𝟐 and g(x) = x+2

Using the remainder theorem, find the remainder, when p(x) is divided by g(x), where

p(x) = 𝒙^𝟑 − 𝟔𝒙^𝟐 + 𝟗𝒙 + 𝟑, g(x) = x-1

p(x) = 𝟐𝒙^𝟑 − 𝟕𝒙^𝟐 + 𝟗𝒙 − 𝟏𝟑, g(x) = x-3

p(x) = 𝟑𝒙^𝟒 − 𝟔𝒙^𝟐 − 𝟖𝒙 − 𝟐, g(x) = x-2

p(x) = 2𝒙^𝟑 − 𝟗𝒙^𝟐 + 𝒙 + 𝟏𝟓, g(x) = 2x -3

p(x) = 𝒙^𝟑 − 𝟐𝒙^𝟐 − 𝟖𝒙 − 𝟏, g(x) = x+1

p(x) = 𝟐𝒙^𝟑 + 𝒙^𝟐 − 𝟏𝟓𝒙 − 𝟏𝟐, g(x) = x+2

p(x) = 𝟔𝒙^𝟑 + 𝟏𝟑𝒙^𝟐 + 𝟑, g(x) = 3x+2

p(x) = 𝒙^𝟑 − 𝟔𝒙^𝟐 + 𝟐𝒙 − 𝟒, g(x) = 1 - 𝟑/𝟐x

p(x) = 𝟐𝒙^𝟑 + 𝟑𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟑, g(x) = (x+𝟏/𝟐)

p(x) = 𝒙^𝟑 − 𝒂𝒙^𝟐 + 𝟔𝒙 − 𝒂, g(x) = x-a

The polynomials (𝟐𝒙^𝟑 + 𝒙^𝟐 − 𝒂𝒙 + 𝟐) and (𝟐𝒙^𝟑-3𝒙^𝟐-3x+a) when divided by (x-2) leave the same remainder. Find the value of a.

The polynomial p(x) = 𝒙^𝟒 − 𝟐𝒙^𝟑 + 𝟑𝒙^𝟐 − 𝒂𝒙 + 𝒃 when divided by (x-1) and (x+1) leaves the remainders 5 and 19 respectively. Find the values of ‘a’ and ‘b’. Hence find the remainder when p(x) is divided by (x-2).

If p(x) = 𝒙^𝟑 − 𝟓𝒙^𝟐 + 𝟒𝒙 − 𝟑 and g(x) = x-2, show that p(x) is not a multiple of g(x).

If p(x) = 𝟐𝒙^𝟑 − 𝟏𝟏𝒙^𝟐 − 𝟒𝒙 + 𝟓 and g(x) = 2x+1, show that g(x) is not a factor of p(x).

Questions in Polynomials Exercise 2D

Using factor theorem, show that g(x) is a factor of p(x), when

p(x) = 𝒙^𝟑 − 𝟖, g(x) = x-2

p(x) = 𝟐𝒙^𝟑 + 𝟕𝒙^𝟐 − 𝟐𝟒𝒙 − 𝟒𝟓, g(x) = x-3

p(x) = 𝟐𝒙^𝟒 + 𝟗𝒙^𝟑 + 𝟔𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟔, g(x) = x-1

p(x) = 𝒙^𝟒 − 𝒙^𝟐 − 𝟏𝟐, g(x) = x + 2

p(x) = 69 + 11x - 𝒙^𝟐 + 𝒙^𝟑, g(x) = x+3

p(x) = 𝟐𝒙^𝟑 + 𝟗𝒙^𝟐 − 𝟏𝟏𝒙 − 𝟑𝟎, g(x) = x+5

p(x) = 𝟐𝒙^𝟒 + 𝒙^𝟑 − 𝟖𝒙^𝟐 − 𝒙 + 𝟔, g(x) = 2x – 3

p(x) = 𝟑𝒙^𝟑 + 𝒙^𝟐 − 𝟐𝟎𝒙 + 𝟏𝟐, g(x) = 3x – 2

p(x) = 𝟕𝒙^𝟐 − 𝟒√𝟐𝒙 − 𝟔, g(x) = x - √𝟐

p(x) = 𝟐√𝟐𝒙^𝟐 + 𝟓𝒙 + √𝟐, g(x) = x+√𝟐

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

Find the value of k for which (x-1) is a factor of (𝟐𝒙^𝟑 + 𝟗𝒙^𝟐 + 𝒙 + 𝒌).

Find the value of a for which (x-4) is a factor of (𝟐𝒙^𝟑 − 𝟑𝒙^𝟐 − 𝟏𝟖𝒙 + 𝒂).

Find the value of a for which (x+1) is a factor of (a𝒙^𝟑 + 𝒙^𝟐 − 𝟐𝒙 + 𝟒𝒂 − 𝟗).

Find the value of a for which (x+2a) is a factor of (𝒙^𝟓 − 𝟒𝒂^𝟐𝒙^𝟑 + 𝟐𝒙 + 𝟐𝒂 + 𝟑).

Find the value of m for which (2x-1) is a factor of (8𝒙^𝟒 + 𝟒𝒙^𝟑 − 𝟏𝟔𝒙^𝟐 + 𝟏𝟎𝒙 + 𝒎).

Find the value of ‘a’ for which the polynomial (𝒙^𝟒 − 𝒙^𝟑 − 𝟏𝟏𝒙^𝟐 − 𝒙 + 𝒂)is divisible by (x+3).

Without actual division, show that (𝒙^𝟑 − 𝟑𝒙^𝟐 − 𝟏𝟑𝒙 + 𝟏𝟓)is exactly divisible by (𝒙^𝟐+2x-3).

If (𝒙^𝟑 + 𝒂𝒙^𝟐 + 𝒃𝒙 + 𝟔) has (x-2) as a factor and leaves a remainder 3 when divided by (x-3), find the values of ‘a’ and ‘b’.

Find the values of ‘a’ and ‘b’ so that the polynomial (𝒙^𝟑 − 𝟏𝟎𝒙^𝟐 + 𝒂𝒙 + 𝒃) is exactly divisible by (x-1) as well as (x-2).

Find the values of a and b so that the polynomial (𝒙^𝟒 + 𝒂𝒙^𝟑 − 𝟕𝒙^𝟐 − 𝟖𝒙 + 𝒃) is exactly divisible by (x+2) as well as(x+3).

If both (x-2) and (x - 𝟏/𝟐) are factors of p𝒙^𝟐+ 5x + r, prove that p = r.

Without actual division, prove that 𝟐𝒙^𝟒 − 𝟓𝒙^𝟑 + 𝟐𝒙^𝟐 − 𝒙 + 𝟐 is divisible by 𝒙^𝟐- 3x + 2.

What must be added to 𝟐𝒙^𝟒 − 𝟓𝒙^𝟑 + 𝟐𝒙^𝟐 − 𝒙 − 𝟑 so that the result is exactly divisible by (x-2)?

What must be subtracted from (𝒙^𝟒 + 𝟐𝒙^𝟑 − 𝟐𝒙^𝟐 + 𝟒𝒙 + 𝟔) so that the result is exactly divisible by

(𝒙^𝟐+2x – 3)?

Use factor theorem to prove that (x + a) is a factor of (𝒙^𝒏 + 𝒂^𝒏)for any odd positive integer n.

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Chapters in this book

Number Systems

Polynomials

Factorization of Polynomials

Linear Equations In Two Variables

Coordinate Geometry

Introduction To Euclids Geometry

Lines And Angles

Triangles

Congruence Of Triangles And Inequalities In A Triangle

Quadrilaterals

Areas Of Parallelograms And Triangles

Circles

Geometrical Constructions

Areas Of Triangles And Quadrilaterals

Volume And Surface Area Of Solids

Mean Median And Mode Of Ungrouped Data

Probability