**HCF of Polynomials by Factorization** | **List of top 20 HCF Questions And Answers For Class 10 with polynomials expressions and factors. | HCF Questions and Answers of Polynomials **

HCF is the highest common factor of any two or more than two numbers or polynomials. If we will take 6, 12, and 18 numbers then 6 is the HCF of these numbers.

Let’s see HCF questions and answers for class 10. We will see HCF of Polynomials By Factorization method.

## HCF Questions And Answers For Class 10

**Que.1 Find the Highest Common Factor ( HCF ) of x**^{2} – 9 and x^{2} – 5x + 6.

^{2}– 9 and x

^{2}– 5x + 6.

**Ans:** First of all we will do the factors of these two polynomials.

**x ^{2} – 9** =

**x**3

^{2}–^{2}

= (x+3)(x-3)

**x ^{2} – 5x + 6**

=

**x**

^{2}– 5x + 6=

**x**

^{2}– 3x -2x + 6= x(x-3) -2(x-3)

= (x-2)(x-3)

The factors of **x ^{2} – 9 and x^{2} – 5x + 6** are as follow

**x ^{2} – 9 = **(x+3)(x-3)

**x**= (x-2)(x-3)

^{2}– 5x + 6The common factors of these two polynomials is (x-3)

So, the HCF (Highest Common Factor ) of **x ^{2} – 9 and x^{2} – 5x + 6** is (x-3).

HCF of Polynomials By Factorization is (x-3)

#### Que. 2. Find the HCF of x2 – 4 and x3 + 8.

**Ans:** First of all we will do the factors of these two polynomials.

**x ^{2} – **4 = (

**x)**(2)

^{2}–^{2}

= (x+2)(x-2)

x^{3} + 8 = (x)^{3} + (2)^{3}

= (x+2)(x^{2}-2x+4)

We find that the factors of these two polynomials x2 – 4 and x3 + 8 are as follow:

**x ^{2} – **4 = (x+2)(x-2)

x

^{3}+ 8 = (x+2)(x

^{2}-2x+4)

Here highest common factor is (x+2)

The HCF of x2 – 4 and x3 + 8 is (x+2)

#### Que. 3. Find the highest common factor of (1+x^{3}) and (1-x+x^{2})(1+x+x^{2})

**Ans:** First of all, we will do the factors of these polynomials

**(1+x ^{3})** = (1+x)(1+x-x

^{2})

(1-x+x^{2})(1+x+x^{2}) = (1-x+x^{2})(1+x+x^{2})

(note: here (1-x+x^{2})(1+x+x^{2}) this polynomial is in already factorized form)

The HCF (Highest Common Factors) of these (1+x^{3}) and (1-x+x^{2})(1+x+x^{2}) polynomials is (1-x+x^{2}).

#### Que. 4. Find the Highest Common Factor of 3x^{3} + 81 and 2(x+3)

**Ans: **For finding out the highest common factor of 3x^{3} + 81 and 2(x+3) polynomials. We have to do factors of these two polynomials.

**3x ^{3} + 81 **

In this polynomial first of all we will take 3 common.

3(x^{3} + 27)

= 3(x^{3} + 3^{3})

= 3 (x+3)(x-3x+9)

**2(x+3)**

This polynomial is already in factorized form

From both polynomials factors we can see (x+3) is common.

Hence the highest common factor of these two polynomials 3x^{3} + 81 and 2(x+3) is (x+3).

#### Que. 5. Find the highest common factors of the polynomials (x^{2}+x),(x+1)^{3} and (x^{3}+1)

**Ans. ** All these three polynomials (x^{2}+x),(x+1)^{3} and (x^{3}+1) are very simple to factorized.

In this polynomial (x+1)^{3} We will use a plus b whole cube formula and (x^{3}+1) in this polynomial we will use a^{3} +b^{3} formula.

Let’s do the factors of polynomials ofone by one.

**(x ^{2}+x)** = x(x+1)

**(x+1)**= (x+1)(1+x)(1+x)

^{3}**(x**= (x+1)(1-x+x

^{3}+1)^{2})

After doing the factorization of these polynomials we find (x+1) is the highest common factor.

Hence the HCF of polynomials (x^{2}+x),(x+1)^{3} and (x^{3}+1) is (X + 1).

#### Que. 6. Find the highest common factor x3-27 and x2 + 4x – 21.

Ans. Given polynomials are x^{3}-27 and x^{2} + 4x – 21

The factors of these polynomials x^{3}-27 and x^{2} + 4x – 21 are as follow:**x ^{3}-27 = x^{3}–**3

^{3}

=

**(x-3)(x**

x=

^{2}+3x+9)x

^{2}+ 4x – 21**x**+ 7x – 3x – 21

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

= (x-3)(x+7)

From the factors of both polynomials we can see the highest common factor is (x-3).

#### Que. 7. Find the HCF (highest common factors) of polynomials x3 – 4x, x2 – 4x + 4, and x4 – 16

Ans: in this question we can see the 3 polynomials are given. These polynomials are x3 – 4x, x2 – 4x, and x4 – 16.

First of all we will do the factors of these polynomials.

x^{3} – 4x = x(x^{2}-4) = x(x+2)(x-2)

x^{2} – 4x + 4 =(x-2)(x-2)

x^{4} – 16 = (x^{2})^{2} -4^{2}

=(x^{2}-4)(x^{2}+4)

=(x+2)(x-2)(x^{2}+4)

Here after factorization we noticed that highest common factor is (x-2).

Hence the HCF of Polynomials By Factorization is x3 – 4x, x2 – 4x + 4, and x4 – 16 is (X – 2). And this is our answer.

#### Que. 8. Find the highest common factor of polynomials (1-x^{2}) & (1+x^{3})

Ans. These polynomials (1-x^{2}) & (1+x^{3}) are very simple to factorise.

Main reason behind the factorization is we can put this value in very simple formula.

(1-x^{2}) =a^{2} – b^{2}

(1+x^{3}) =a^{3} + b^{3}

Now we will do the factorization of these two polynomials one by one.

The factors of (1-x^{2}) are (1-x)(1+x)

The factors of (1+x^{3}) are (1+x)(1+x^{2}-x)

From the factors of these two polynomials, we can see that the highest common factor is (1+x).

Hence the HCF of polynomials (1-x^{2}) & (1+x^{3}) is (1+x).

#### Que. 9. Find the highest common factor of 9-p^{2} and 27 – p^{3}

After seeing the polynomials 9-p^{2} and 27 – p^{3} We can understand that these are the standard formulas.

So for doing the the factors of these polynomials we have to factorise them into formula expression.

Let’s do the factorization of these two polynomials one by one.

The factors of 9-p^{2} are as follow:

3^{2}-p^{2} = (3-p)(3+p)

The factors of 27 – p^{3} are as follow:

27 – p^{3} = 3^{3} – p^{3}

= (3-p)(9 + p^{2} +3p)

The common factor From the factorization of polynomials 9-p^{2} and 27 – p^{3} Is (3-p)

HCF of the polynomials 9-p^{2} and 27 – p^{3} Is (3-p).

#### Que. 10. Find the highest common factor of x^{2} – 9 and x^{3} + 27

After seeing the polynomials x^{2} – 9 and x^{3} + 27 We can understand that these are the standard formulas.

So for doing the the factors of these polynomials we have to factorise them into formula expression.

Let’s do the factorization of these two polynomials one by one.

The factors of x^{2} – 9 are as follow:

x^{2} – 9 = (x – 3)(x + 3)

The factors of x^{3} + 27 are as follow:

x^{3} + 27 = x^{3} + 3^{3}

= (x+3)(9 + x^{2} -3x)

The common factor From the factorization of polynomials x^{2} – 9 and x^{3} + 27 is (x+3)

HCF of the polynomials x^{2} – 9 and x^{3} + 27 Is (x + 3).

## HCF of Polynomials By Factorization Questions And Answers

#### Que. 11. Find the highest common factor of Polynomials 8(x^{3}– x^{2} +x) and 28(x^{3} + 1)

Ans. First of all we will do the factors of these polynomials 8(x^{3} – x^{2} +x) and 28(x^{3} + 1)

The factors of polynomial 8(x^{3} – x^{2} +x) are as follow:

8(x^{3} – x^{2} +x) = 8x(x^{2}-x+1)

= (2*2*2x)(x^{2}-x+1)

and

The factors of polynomial 28(x^{3} + 1) are as follow:

28(x^{3} + 1) = 28(x^{3} + 1)

= 2*2*7(x+1)(1-x+x^{2})

Here we can see that the common factors of polynomials 8(x^{3}– x^{2} +x) and 28(x^{3} + 1) are 2*2*(1-x+x^{2})

Hence the HCF (Highest Common Factor) of polynomials 8(x^{3}– x^{2} +x) and 28(x^{3} + 1) is **4(1-x+x ^{2})**.

#### Que. 12. Find the Highest Common Factors of Polynomials (x+2)^{2} and (3x^{2}-12)

As we all know for doing the factorization of Polynomials (x+2)^{2} and (3x^{2}-12). We have to factorize that.

The Factors of the Polynomial (x+2)^{2} are as follow:

(x+2)^{2} = (x+2)(x+2)

and

The Factors of the Polynomial (3x^{2}-12) are as follow:

(3x^{2}-12) = 3(x^{2}-4)

= 3(x-2)(x+2)

Here we can see that the common factors of polynomials (x+2)^{2} and (3x^{2}-12) is (x + 2)

Hence the HCF of Polynomials By Factorization (Highest Common Factor) of (x+2)^{2} and (3x^{2}-12) is (x + 2).

#### Que. 13. Find the Highest Common Factors of Polynomials x^{2} – 6x + 9 and x^{2} – 9

As we all know for doing the factorization of Polynomials x^{2} – 6x + 9 and x^{2} – 9. We have to factorize that.

The factors of the polynomial x^{2} – 6x + 9 are as follow:

x^{2} – 6x + 9 = x^{2} – 3x – 3x + 9

= x(x-3) -3(x-3)

=(x-3)(x-3)

The factors of the polynomial (x^{2} – 9) are as follow:

x^{2} – 9 = (x – 3)(x + 3)

Here we can see that the common factors of polynomials (x^{2} – 6x + 9) and (x^{2} – 9) is (x-3)

Hence the HCF (Highest Common Factor) of polynomials (x^{2} – 6x + 9) and (x^{2} – 9) is (x-3)

#### Que. 14. Find the Highest Common Factors HCF of Polynomials (x^{2}-2xy+y^{2}) and (x^{3}y-xy^{3})

Ans. As we all know for doing the factorization of Polynomials (x^{2}-2xy+y^{2}) and (x^{3}y-xy^{3}). We have to factorize that.

The factors of the polynomial (x^{2}-2xy+y^{2}) are as follow:

here (x^{2}-2xy+y^{2}) is seeing difficult to factorize as it shows two variables values x and y. But you don’t worry. Just assume y as constant value like number and then solve this equation.

(x^{2}-2xy+y^{2})

= (x^{2}-xy -xy +y^{2})

= x(x-y) -y(x-y)

= (x-y)(x-y)

The factors of the polynomial (x^{3}y-xy^{3}) are as follow:

(x^{3}y-xy^{3})

This is very simple to slove.

first of all we will take xy common from this polynomial.

x^{3}y-xy^{3}

= xy(x^{2} – y^{2})

= xy(x-y)(x+y)

Here we can see that the common factors of polynomials (x^{2}-2xy+y^{2}) and (x^{3}y-xy^{3}) are (x-y)

Hence the HCF (Highest Common Factor) of polynomials (x^{2}-2xy+y^{2}) and (x^{3}y-xy^{3}) is (x-y).

#### Que. 15. Find the Highest Common Factors HCF of Polynomials (x^{2}– 4) and (3x^{3}+ 24).

Ans. As we all know for doing the factorization of Polynomials (x^{2}– 4) and (3x^{3}+ 24). We have to factorize that.

The factors of the polynomial (x^{2}– 4) are as follow:

x^{2}– 4

= (x^{2}-2^{2})

= (x-2)(x+2)

The factors of the polynomial (3x^{3}+ 24) are as follow:

(3x^{3}+ 24)

= 3(x^{3}+ 8)

= 3(x^{3} + 2^{3})

= 3(x+2)(x^{2} + 4 -2x)

Here we can see that the common factor of polynomials (x^{2}– 4) and (3x^{3}+ 24) is (x + 2)

Hence the HCF of Polynomials By Factorization (Highest Common Factor) of (x^{2}– 4) and (3x^{3}+ 24) is (x + 2).

#### Que. 16. Find the HCF of polynomials 2x^{2}y(x^{2} – y^{2}) and 4xy^{2}(x-y)

Ans. In this question factorization is very simple. here we have to treat one variable as constant number. Then every equation become easy to enter value from the algebra formula.

Let’s do the factorization of polynomials 2x^{2}y(x^{2} – y^{2}) and 4xy^{2}(x-y) for finding the HCF.

The factors of the polynomial 2x^{2}y(x^{2} – y^{2}) are as follow:

2x^{2}y(x^{2} – y^{2})

= 2x^{2}y(x + y)(x – y)

and

The factors of the polynomial 4xy^{2}(x-y) are as follow:

4xy^{2}(x-y)

= 2*2xy^{2}(x-y)

Here we can see that the common factor of polynomials 2x^{2}y(x^{2} – y^{2}) and 4xy^{2}(x-y) are 2xy(x – y).

Hence the HCF (Highest Common Factor) of polynomials 2x^{2}y(x^{2} – y^{2}) and 4xy^{2}(x-y) are 2xy(x – y).

#### Que. 17. Find the Highest Common Factors of the polynomial (x^{2} – x – 6) & (x^{3} – 27).

For finding out the HCF of polynomials (x^{2} – x – 6) & (x^{3} – 27), we have to factorized them.

The factors of the polynomial (x^{2} – x – 6) are as follow:

(x^{2} – x – 6)

= x^{2} – 3x + 2x – 6

= x(x-3)+2(x-3)

(x+2)(x-3)

And

The factors of the polynomial (x^{3} – 27) are as follow:

x^{3} – 27

= x^{3} – 3^{3}

= (x-3)(x^{2}+3x+9)

The common factor of these two polynomials (x^{2} – x – 6) & (x^{3} – 27) is (x-3)

Hence The HCF of the polynomials (x^{2} – x – 6) & (x^{3} – 27) is (x-3).

#### Que. 18. Find the Highest Common Factor of the polynomial (x^{3}-1) and (x^{4} + x^{2} + 1).

Ans. In these polynomials (x^{3}-1) and (x^{4} + x^{2} + 1). (x^{4} + x^{2} + 1) has the fourth power of x. So, it may confused you. But don’t worry. This is a very simple question. In this, we will assume x^{2} = X. And then exuation is very simple in quadratic format.

So, lets do the factorization:

The factors of the polynomial (x^{3}-1) are as follow:

(x^{3}-1)

= x^{3}-1

= (x-1)(x^{2}+1+x)

The factors of the polynomial (x^{4} + x^{2} + 1) are as follow:

(x^{4} + x^{2} + 1)

= (x^{2})^{2} + x^{2} + 1

= (x^{2})^{2} + (x^{2}) +1 +2

(x^{4} + x^{2} + 1)

= (x^{2})^{2} + x^{2} + 1

= (x^{2})^{2} +1 +2x^{2} – x^{2}

= (x^{2} + 1)^{2} – x^{2}

= (x^{2}+ 1 + x)(x^{2} + 1 – x)

(Note: here we try to make a formula of a^{2} – b^{2} because polynomial x^{4} + x^{2} + 1 can’t be factorized be square root method)

The common factor of these two polynomials (x^{3}-1) and (x^{4} + x^{2} + 1) is (x^{2}+ 1 + x).

Hence The HCF of the polynomials (x^{3}-1) and (x^{4} + x^{2} + 1) is (x^{2}+ 1 + x).

#### Que. 19. Find the highest common factors of the polynomial 3ax^{2}(x^{2}-a^{2}) and 6ax(x^{3}+a^{3}).

Ans. First of all we will do the factorization of these both polynomials 3ax^{2}(x^{2}-a^{2}) and 6ax(x^{3}+a^{3}). And then we will find the highest common factor and then we will decide the HCF.

The factors of the polynomial 3ax^{2}(x^{2}-a^{2}) are as follow:

3ax^{2}(x^{2}-a^{2})

= 3ax^{2}(x+a)(x-a)

and

The factors of the polynomial 6ax(x^{3}+a^{3}) are as follow:

6ax(x^{3}+a^{3})

= 2*3ax(x+a)(x^{2}+a^{2}-ax)

Here we can find that 3ax(x+a) is the highest factor that can divide polynomials 3ax^{2}(x^{2}-a^{2}) and 6ax(x^{3}+a^{3}).

Hence the highest common factor HCF of Polynomials By Factorization 3ax^{2}(x^{2}-a^{2}) and 6ax(x^{3}+a^{3}) is 3ax(x+a).

#### Que. 20. Find the highest common factors of the polynomials 2x^{3} – x^{2} – x and 4x^{2} + 8x + 3.

For finding out the HCF we will do factors of these polynomials 2x^{3} – x^{2} – x and 4x^{2} + 8x + 3.

Factors of the polynomial 2x^{3} – x^{2} – x are as follow:

2x^{3} – x^{2} – x

= x(2x^{2} – x – 1)

= x(2x^{2} – 2x + x – 1)

= x[2x(x-1)+1(x-1)]

= x(2x+1)(x-1)

and

Factors of the polynomial 4x^{2} + 8x + 3 are as follow:

4x^{2} + 8x + 3

= 4x^{2} + 6x + 2x + 3

= 2x(2x+3)+1(2x+3)

= (2x+1)(2x+3)

Here we can find that (2x+1) is the highest factor that can divide polynomials 2x^{3} – x^{2} – x and 4x^{2} + 8x + 3.

Hence the highest common factor HCF of Polynomials By Factorization is (2x^{3} – x^{2} – x) and (4x^{2} + 8x + 3) is (2x+1).

Also, Read:

- Find The Zeros of The Polynomial p(x)=(x-2)^2-(x+2)^2.
- 8×2-22x-21=0 By Factorization Method
- Find The Zeros of The Quadratic Polynomial x2-2x-8 = 0
- Factorization of 3×2-8x+5

In this article, we learned HCF of Polynomials By Factorization questions and answers for class 10 NCERT and UP Board.

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