Nootan solutions for माठेमटिक्स [इंग्रजी] इयत्ता १० आयसीएसई chapter 6 - Factorisation of polynomials [Latest edition]

Find the remainder of the following case when f(x) is divided by g(x).

f(x) = x² + 6x + 10, g(x) = x − 3

Find the remainder of the following case when f(x) is divided by g(x).

f(x) = 4x² − 5x + 1, g(x) = 2x − 1

Find the remainder of the following case when f(x) is divided by g(x).

f(x) = x³ + 4x² − 6x + 8, g(x) = x + 1

Find the remainder of the following case when f(x) is divided by g(x).

f(x) = 3x³ + 4x² − 7, g(x) = x + 2

Find the value of k if the remainder is 12 when x³ + 4x² − 7x + k is divided by x − 2.

Find the value of k if the remainder is −6 when 2x³ − 6x² + kx + 6 is divided by x + 3.

Determine whether g(x) is a factor of f(x) or not:

f(x) = x² − 7x + 6; g(x) = x − 6

Determine whether g(x) is a factor of f(x) or not:

f(x) = 4x² + 7x − 3; g(x) = 2x + 3

Determine whether g(x) is a factor of f(x) or not:

f(x) = x³ − x² − x − 2; g(x) = x − 2

Determine whether g(x) is a factor of f(x) or not:

f(x) = 5x³ + x − 6; g(x) = x − 1

Determine whether g(x) is a factor of f(x) or not:

f(x) = 8x³ + 4x² − 6x − 36; g(x) = x + 2

Determine whether g(x) is a factor of f(x) or not:

f(x) = 6x³ − x + 3; g(x) = 2x + 1

Check whether 7 + 3x is a factor of 3x³ + 7x.

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

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 the following cases, if 2R₁ − R₂ = 0.

Find the value of k, if x – 1 is a factor of p(x) in the following case:

p(x) = x² + x + k

In the following two polynomials. Find the value of ‘a’ if x + a is a factor of each of the two:

x³ + ax² − 2x + a + 4

Find the value of k if (x – 2) is a factor of 2x³ – 6x² + 5x + k. Also find whether x – 1 is a factor or not.

If x + 1 and x − 1 are the factors of px³ + x² − 2x + q, find the value of p and q.

If the polynomial x³ + ax² − bx − 6 is exactly divisible by (x² − x − 2), find the values of a and b.

Find the values of p and q if (x + 1) and (x + 2) are the factors of x³ + px² + 11x + q.

If x − 2 is a factor of x³ + px² + qx − 4 and when the expression is divided by x − 3, it leaves a remainder 17, find the values of p and q.

What number must be added to x³ + 5x² − 6x so that the resulting polynomial is exactly divisible by (x – 2)?

What number must be added to 2x³ + x² + 17x + 10 so that the resulting polynomial is exactly divisible by (x + 2)?

What number must be subtracted from the polynomial x³ − 6x² + 2x so that (x – 1) is a factor of resulting polynomial?

What number must be subtracted from the polynomial 4x³ − 6x² + x + 17 so that when the resulting polynomial is divided by x − 3, it leaves a remainder 10?

Factorise using factor theorem:

x³ + 13x² + 31х − 45

Factorise using factor theorem:

x³ − 23x² + 142x − 120

Factorise using factor theorem:

3x³ − 4x² − 12x + 16

Factorise using factor theorem:

x³ − 7x + 6

Prove that (x − 2) is a factor of x³ − 7x² + 14x − 8. Hence, completely factorise the expression.

Prove that (x + 3) is a factor of x³ − 2x² − 9x + 18. Hence, factorise it completely.

Prove that (2x + 1) is a factor of 2x³ + x² − 8x − 4. Hence, factorise it completely.

Prove that (3x + 2) is a factor of 3x³ + 5x² − 4x − 4. Hence, factorise it completely.

If (x + 2) is a factor of x³ + 2x² + kx − 18, find the value of k. Hence, factorise it completely.

If (2x + 1) is a factor of 2x³ − 9x² + kx + 6, find the value of k. Hence, factorise it completely.

If (x − 2) is a factor of 2x³ − x² − px − 2, then:

1. Find the value of p.
2. With the value of p, factorise the above expression completely.

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.

The polynomial 3x³ + 8x² – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.

When x³ − 2x² + 6x − k is divided by x − 2, the remainder is −10. The value of k is ______.

If 3x² + kx − 2 is exactly divisible by x − 1, then the value of k is ______.

Which one is the factor of 2x² − 7x + 3?

When x³ +x² + x + 1 is divided by x + 2, the remainder is ______.

If x + 3 and 2x + 1 are the factors of 6x³ + px² + qx − 6, then the value of p is ______.

When the polynomials x³ − px² + x + 6 and 2x³ − x² − (p + 3) x − 6 are divided by x − 3, they leave the same remainder, the value of p is ______.

The factors of 3x³ − 7x² + 4 are ______.

If 2x + 1 is a factor of 3 − kx − 2x², then the value of k is ______.

If x − 4 is a factor of x³ + kx² + 2x − 8, then the value of k is ______.

If x − a is a factor of 2x³ − 2a²x + x − 5, then the value of ‘a’ is ______.

What must be subtracted from the polynomial x³ + x² – 2x + 1, so that the result is exactly divisible by (x – 3)?

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. When the polynomial f(x) = 4x³ − 7x + 3 is divided by x − 1, the remainder is 0.
2. x − a is a factor of the polynomial f(x) if f(a) = 0.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If x − a is a factor of 3x³ + x² − ax − 81, then a = 3.
2. x + 1 is a factor of 3x³ − 7x² + 4.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. 4x² + kx − 7 is exactly divisible by x − 2 then the value of k is `9/2`.
2. (2x + 1) is a factor of 2x³ + x² − 8x − 4.

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. If (x + 1) and (x + 2) are the factors of x³ + px² + qx − 2 then p = 0.
2. 2x³ + x² + 17x + 46 is exactly divisible by x + 2.

Use factor theorem to factorise: 3x³ + 2x² − 19x + 6.

Determine whether x + 1 is a factor of x³ + x² − 8x − 8.

Use factor theorem to factorise 6x³ + 17x² + 4x − 12.

What must be added to the polynomial 2x³ − 3x² − 8x so that it leaves a remainder 10 when divided by (2x + 1)?

Using the remainder theorem, find the remainders obtained when x³ + (kx + 8 )x + k is divided by x + 1 and x − 2. 

Hence, find k if the sum of the two remainders is 1.

If the polynomial 2x³ + 3x² + px + q is exactly divisible by 2x² + 9x + 9, find the value of p and q. Hence, factorise the expression completely.

The polynomials 5x³ − 3x² + kx − 25 and x³ + 15x + 2k + 18  leave the same remainder when divided by x − 3. Find the value of k.