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.
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