Selina solutions for Mathematics [English] Class 10 ICSE chapter 8 - Remainder and Factor Theorems [Latest edition]

Find the remainder when x⁴ – 3x² + 2x + 1 is divided by x – 1.

Find the remainder when x³ + 3x² – 12x + 4 is divided by x – 2.

Find the remainder when x⁴ + 1 is divided by x + 1.

Show that x – 2 is a factor of 5x² + 15x – 50.

Show that 3x + 2 is a factor of 3x² – x – 2.

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

x + 1

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

2x – 1 

Use the Remainder Theorem to find which of the following is a factor of 2x³ + 3x² – 5x – 6. 

x + 2

If 2x + 1 is a factor of 2x² + ax – 3, find the value of a.

Find the value of k, if 3x – 4 is a factor of expression 3x² + 2x − k.

Find the values of constants a and b when x – 2 and x + 3 both are the factors of expression x³ + ax² + bx – 12.

Find the value of k, if 2x + 1 is a factor of (3k + 2)x³ + (k − 1)

Find the value of a, if x – 2 is a factor of 2x⁵ – 6x⁴ – 2ax³ + 6ax² + 4ax + 8. 

Find the values of m and n so that x – 1 and x + 2 both are factors of x³ + (3m + 1)x² + nx – 18.

When x³ + 2x² – kx + 4 is divided by x – 2, the remainder is k. Find the value of constant k.

Find the value of a, if the division of ax³ + 9x² + 4x – 10 by x + 3 leaves a remainder 5.

If x³ + ax² + 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.

The expression 2x³ + ax² + bx – 2 leaves remainder 7 and 0 when divided by 2x – 3 and x + 2 respectively. Calculate the values of a and b.

What number should be added to 3x³ – 5x² + 6x so that when resulting polynomial is divided by x – 3, the remainder is 8?

What number should be subtracted from x³ + 3x² – 8x + 14 so that on dividing it by x – 2, the remainder is 10? 

The polynomials 2x³ – 7x² + ax – 6 and x³ – 8x² + (2a + 1)x – 16 leaves the same remainder when divided by x – 2. Find the value of ‘a’. 

If (x – 2) is a factor of the expression 2x³ + ax² + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b.

Find ‘a‘ if the two polynomials ax³ + 3x² – 9 and 2x³ + 4x + a, leave the same remainder when divided by x + 3. 

Using the Factor Theorem, show that (x – 2) is a factor of x³ – 2x² – 9x + 18. Hence, factorise the expression x³ – 2x² – 9x + 18 completely.

Using the Factor Theorem, show that (x + 5) is a factor of 2x³ + 5x² – 28x – 15. Hence, factorise the expression 2x³ + 5x² – 28x – 15 completely.  

Using the Factor Theorem, show that (3x + 2) is a factor of 3x³ + 2x² – 3x – 2. Hence, factorise the expression 3x³ + 2x² – 3x – 2 completely.

Using the Remainder Theorem, factorise each of the following completely. 

3x³ + 2x² − 19x + 6 

Using the Reminder Theorem, factorise of the following completely.

2x³ + x² – 13x + 6

Using the Remainder Theorem, factorise the following completely:

3x³ + 2x² – 23x – 30

Using the Remainder Theorem, factorise the following completely:  

4x³ + 7x² – 36x – 63

Using the Remainder Theorem, factorise the following completely:

x³ + x² – 4x – 4

Using the Remainder Theorem, factorise the expression 3x³ + 10x² + x – 6. Hence, solve the equation 3x³ + 10x² + x – 6 = 0

Factorise the expression f(x) = 2x³ – 7x² – 3x + 18. Hence, find all possible values of x for which f(x) = 0.

Given that x – 2 and x + 1 are factors of f(x) = x³ + 3x² + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

The expression 4x³ – bx² + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely. 

If x + a is a common factor of expressions f(x) = x² + px + q and g(x) = x² + mx + n; show that : ${\displaystyle a=\frac{n-q}{m-p}}$ 

The polynomials ax³ + 3x² – 3 and 2x³ – 5x + a, when divided by x – 4, leave the same remainder in each case. Find the value of a.

Find the value of ‘a’, if (x – a) is a factor of x³ – ax² + x + 2.

Find the number that must be subtracted from the polynomial 3y³ + y² – 22y + 15, so that the resulting polynomial is completely divisible by y + 3. 

Show that (x – 1) is a factor of x³ – 7x² + 14x – 8. Hence, completely factorise the given expression.

Using Remainder Theorem, factorise : x³ + 10x² – 37x + 26 completely.

When x³ + 3x² – mx + 4 is divided by x – 2, the remainder is m + 3. Find the value of m.

What should be subtracted from 3x³ – 8x² + 4x – 3, so that the resulting expression has x + 2 as a factor?

If (x + 1) and (x – 2) are factors of x³ + (a + 1)x² – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.

If x – 2 is a factor of x² + ax + b and a + b = 1, find the values of a and b.

Factorise x³ + 6x² + 11x + 6 completely using factor theorem. 

Find the value of ‘m’, if mx³ + 2x² – 3 and x² – mx + 4 leave the same remainder when each is divided by x – 2.

The polynomial px³ + 4x² – 3x + q is completely divisible by x² – 1; find the values of p and q. Also, for these values of p and q, factorize the given polynomial completely.

Find the number which should be added to x² + x + 3 so that the resulting polynomial is completely divisible by (x + 3).

When the polynomial x³ + 2x² – 5ax – 7 is divided by (x – 1), the remainder is A and when the polynomial x³ + ax² – 12x + 16 is divided by (x + 2), the remainder is B. Find the value of ‘a’ if 2A + B = 0.

(3x + 5) is a factor of the polynomial (a – 1)x³ + (a + 1)x² – (2a + 1)x – 15. Find the value of ‘a’, factorise the given polynomial completely.

When divided by x – 3 the polynomials x³ – px² + x + 6 and 2x³ – x² – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.

Using the Remainder Theorem, factorise the following completely:

2x³ + x² – 13x + 6 

Using remainder theorem, find the value of k if on dividing 2x³ + 3x² – kx + 5 by x – 2, leaves a remainder 7.

What must be subtracted from 16x³ – 8x² + 4x + 7 so that the resulting expression has 2x + 1 as a factor?