RD Sharma 2020 solution class 9 chapter 6 Factorization of polynomial Expressions Exercise 6.3

Exercise 6.3

Page-6.14

Question 1:

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x) and verify the result by actual division: (1−8)

1. f(x) = x³ + 4x² − 3x + 10, g(x) = x + 4

Answer 1:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Now we will show by actual division

So the remainder by actual division is 22

Question 2:

f(x) = 4x⁴ − 3x³ − 2x² + x − 7, g(x) = x − 1

Answer 2:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Now we will show remainder by actual division

So the remainder by actual division is −7

Question 3:

f(x) = 2x⁴ − 6x³ + 2x² − x + 2, g(x) = x + 2

Answer 3:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Now we will calculate the remainder by actual division

So the remainder by actual division is 92

Question 4:

f(x) = 4x³ − 12x² + 14x − 3, g(x) 2x − 1

Answer 4:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Now we will calculate the remainder by actual division

So the remainder by actual division is

Question 5:

f(x) = x³ − 6x² + 2x − 4, g(x) = 1 − 2x

Answer 5:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Now we will calculate remainder by actual division

So the remainder is

Question 6:

f(x) = x⁴ − 3x² + 4, g(x) = x − 2

Answer 6:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

We will calculate remainder by actual division

So the remainder is 8

Question 7:

 f(x) = 9x³ − 3x² + x − 5, g(x) = $x-\frac{2}{3}$

Answer 7:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Remainder by actual division

Remainder is −3

Question 8:

$f\left(x\right) = 3x^4 + 2x^3-\frac{x^2}{3}-\frac{x}{9}+\frac{2}{27}, g\left(x\right)=x+\frac{2}{3}$

Answer 8:

Let us denote the given polynomials as

We have to find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Remainder by actual division

Remainder is 0

Question 9:

If the polynomials 2x³ + ax² + 3x − 5 and x³ + x² − 4x +a leave the same remainder when divided by x −2, find the value of a.

Answer 9:

Let us denote the given polynomials as

Now, we will find the remaindersandwhenandrespectively are divided by.

By the remainder theorem, whenis divided bythe remainder is

By the remainder theorem, whenis divided bythe remainder is

By the given condition, the two remainders are same. Then we have,

Question 10:

If the polynomials ax³ + 3x² − 13 and 2x³ − 5x + a, when divided by (x − 2) leave the same remainder, find the value of a.

Answer 10:

Let us denote the given polynomials as

Now, we will find the remaindersandwhenandrespectively are divided by.

By the remainder theorem, whenis divided bythe remainder is

By the remainder theorem, whenis divided bythe remainder is

By the given condition, the two remainders are same. Then we have, R₁ = R₂

Question 11:

Find the remainder when x³ + 3x² + 3x + 1 is divided by

(i) x + 1

(ii) $x-\frac{1}{2}$

(iii) x

(iv) $x+\mathrm{\pi }$

(v) 5 + 2x
 

Answer 11:

Let us denote the given polynomials as

(i) We will find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

(ii) We will find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

(iii) We will find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

(iv) We will find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

(v) We will find the remainder whenis divided by.

By the remainder theorem, whenis divided bythe remainder is

Page-6.15

Question 12:

The polynomials ax³ + 3x² − 3 and 2x³ − 5x + a when divided by (x − 4) leave the remainders R₁ and R₂ respectively. Find the values of a in each of the following cases, if

(a) R₁ = R₂

(b) R₁ + R₂ = 0

(c) 2R₁ − R₂ = 0

Answer 12:

Let us denote the given polynomials as

Now, we will find the remaindersandwhenandrespectively are divided by.

By the remainder theorem, whenis divided bythe remainder is

By the remainder theorem, whenis divided bythe remainder is

(i) By the given condition,

R₁ = R₂

(ii) By the given condition,

R₁ + R₂ = 0

(iii) By the given condition,

2R₁ − R₂ = 0