M l aggarwal understanding icse mathematics class 10 Solutions for Chapter 5 factorization

Exercises in Chapter 5 factorization Grade 10

Factorization Exercise 6

Questions in Factorization Exercise 6

Find the remainder (without division) on dividing f(x) by (x - 2) where

(i) $f(x)=5 x^{2}-7 x+4$

(ii) $f(x)=2 x^{3}-7 x^{2}+3$

Find the remainder when $2 x^{3}-3 x^{2}+4 x+7$ is divided by

(i) x – 2

(ii) x + 3

(iii) 2x + 1

When $2 x^{3}-9 x^{2}+10 x-p$ is divided by (x + 1), the remainder is – 24. Find the value of p.

Using the remainder theorem, find the remainder on dividing f(x) by (x + 3) where

(i) $f(x)=2 x^{2}-5 x+1$

(ii) $f(x)=3 x^{3}+7 x^{2}-5 x+1$

Find the remainder (without division) on dividing f(x) by (2x + 1) where,

(i) $f(x)=4 x^{2}+5 x+3$

(ii) $f(x)=3 x^{3}-7 x^{2}+4 x+11$

If (2x – 3) is a factor of $6 x^{2}+x+a$ , find the value of a. With this value of a, factorize the given expression.

(i) find the remainder (without division) when $2 x^{3}-3 x^{2}+7 x-8$ is divided by x – 1.

(ii) Find the remainder (without division) on dividing $3 x^{2}+5 x-9$ by (3x + 2).

When $3 x^{2}-5 x+p$ is divided by (x – 2), the remainder is 3. Find the value of p. Also factorize the polynomial $3 x^{2}-5 x+p-3$ .

Using the remainder theorem, find the value of k if on dividing $2 x^{3}+3 x^{2}-k x+5$ by x – 2 leaves a remainder 7.

Prove that (5x + 4) is a factor of $5 x^{3}+4 x^{2}-5 x-4$ . Hence factorize the given polynomial completely.

Use factor theorem to factorize the following polynomials completely:

(i) $4 x^{3}+4 x^{2}-9 x-9$

(ii) $x^{3}-19 x-30$

Using the remainder theorem, find the value of ‘a’ if the division of $x^{3}+5 x^{2}-a x+6$ by (x - 1) leaves the remainder 2a.

(i)What number must be divided be subtracted from $2 x^{2}-5 x$ so that the resulting polynomial leaves the remainder 2 when divided by 2x + 1?

(ii) What number must be added to $2 x^{3}-7 x^{2}+2 x$ so that the resulting polynomial leaves the remainder – 2 when divided by 2x – 3?

If $x^{3}-2 x^{2}+p x+q$ has a factor (x + 2) and leaves a remainder 9, when divided by (x + 1), find the values of p and q. With these values of p and q, factorize the given polynomial completely.

(i) When divided by x – 3 the polynomials $x^{3}-p x^{2}+x+6 \text { and } 2 x^{3}-x^{2}-(p+3) x-6$ leave the same remainder. Find the value of ‘p’.

(ii) Find ‘a’ if the two polynomials $a x^{3}+3 x^{2}-9 \text { and } 2 x^{3}+4 x+a$ , leaves the same remainder when divided by x + 3.

If (x + 3) and (x – 4) are factors of $x^{3}+a x^{2}-b x+24$ , find the values of a and b: With these values of a and b, factorize the given expression.

If $2 x^{3}+a x^{2}-11 x+b$ leaves remainder 0 and 42 when divided by (x – 2) and (x – 3) respectively, find the values of a and b. With these values of a and b, factorize the given expression.

By factor theorem, show that (x + 3) and (2x - 1) are factors of $2 x^{2}+5 x-3$ .

If (2x + 1) is a factor of both the expressions $2 x^{2}-5 x+p$ and $2 x^{2}+5 x+9$ , find the value of p and q. Hence find the other factors of both the polynomials.

Without actual division, prove that $x^{4}+2 x^{3}-2 x^{2}+2 x+3$ is exactly divisible by $x^{2}+$ $2 x-3$ .

If a polynomial $f(x)=x^{4}-2 x^{3}+3 x^{2}-a x+b$ leaves reminder 5 and 19 when divided by (x-1) and (x+1) respectively, Find the values of a and b. Hence determine the remainder when f(x) is divided by (x-2).

Show that (x - 2) is a factor of $3 x^{2}-x-10$ . Hence factorise $3 x^{2}-x-10$ .

(i) Show that, (x - 1) is a factor of $x^{3}-5 x^{2}-x+5$ hence factorise $x^{3}-5 x^{2}-x+5$ .

(ii) Show that (x - 3) is a factor of $x^{3}-7 x^{2}+15 x-9$ hence factorise $x^{3}-7 x^{2}+15 x-9$ .

When a polynomial f(x) is divided by (x – 1), the remainder is 5, and when it is, divided by (x – 2), the remainder is 7. Find the remainder when it is divided by (x – 1) (x – 2).

Show that (2x + 1) is a factor of $4 x^{3}+12 x^{2}+11 x+3$ . Hence factorize $-4 x^{3}+12 x^{2}+$ $11 x+3$ .

Show that 2x + 7 is a factor of $2 x^{3}+5 x^{2}-11 x-14$ . Hence factorize the given expression completely, using the factor theorem.

Use factor theorem to factorize the following polynomials completely.

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

(ii) $x^{3}-13 x-12$

Use the remainder theorem to factorize the following expression.

(i) $2 x^{3}+x^{2}-13 x+6$

(ii) $3 x^{2}+2 x^{2}-19 x+6$

(iii) $2 x^{3}+3 x^{2}-9 x-10$

using the remainder and factor theorem factorize the following polynomial $x^{3}+10 x^{2}-37 x+26$

If (2x + 1) is a factor of $6 x^{3}+5 x^{2}+a x-2$ find the value of a.

If (3x - 2) is a factor of $3 x^{3}-k x^{2}+21 x-10$ , find the value of k.

If (x – 2) is a factor of $2 x^{3}-x^{2}+p x-2$ , then (i) find the value of p. (ii) with this

value of p, factorize the above expression completely.

Find the value of ‘K’ for which x = 3 is a solution of the quadratic equation, $\text { (K + }$ $2) x^{2}-k x+6=0$ . Also, find the other root of the equation.

What number should be subtracted from $2 x^{3}-5 x^{2}+5 x$ so that the resulting

the polynomial has 2x – 3 as a factor?

(i) Find the value of the constants a and b, if (x – 2) and (x + 3) are both factors of the expression $x^{3}+a x^{2}+b x-12$ .

(ii) If (x + 2) and (x + 3) are factors of $x^{3}+a x+b$ , Find the values of a and b.

If (x + 2) and (x – 3) are factors of $x^{3}+a x+b$ , find the values of a and b. With these values of a and b, factorize the given expression.

(x – 2) is a factor of the expression $x^{3}+a x^{2}+b x+6$ . When this expression is divided by (x – 3), it leaves the remainder 3. Find the values of a and b.

If (x – 2) is a factor of the expression $2 x^{3}+a x^{2}+b x-14$ and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

If $a x^{3}+3 x^{2}+b x-3$ has a factor (2x + 3) and leaves remainder – 3 when divided by (x + 2), find the values of a and b. With these values of a and b, factorize the given expression.

Given $f(x)=a x^{2}+b x+2 \text { and } g(x)=b x^{2}+a x+1$ . If x – 2 is a factor of f(x) but leaves the remainder – 15 when it divides g(x), find the values of a and b. With these values of a and b, factorise the expression $f(x)+g(x)+4 x^{2}+7 x$ .