In mathematics, factoring is a way to unveil the two or more polynomials that were
multiplied together to obtain the given polynomial. When two or more
polynomials are multiplied together, each one is called a factor of the
resulting product. Factors of polynomials may also be called divisors. The distributive property is used to multiply factors of a polynomial.

Factoring polynomial expressions is not same as factoring numbers, but the concept is similar. When factoring numbers or factoring polynomials, finding numbers or polynomials that divide out evenly from the original numbers or polynomials.

If a polynomial F(x) is the product of two polynomials G(x) and H(x), then each of G(x) and H(x) is called a factor of F(x).

=> F(x) = G(x) * H(x)

Factoring polynomial expressions is not same as factoring numbers, but the concept is similar. When factoring numbers or factoring polynomials, finding numbers or polynomials that divide out evenly from the original numbers or polynomials.

If a polynomial F(x) is the product of two polynomials G(x) and H(x), then each of G(x) and H(x) is called a factor of F(x).

=> F(x) = G(x) * H(x)

## Factoring Polynomial Expressions

### Solved Examples

**Question 1:**Find the factors of 4x

^{3}+ 20x

^{2}+ 8x

**Solution:**

Step 1:

The GCF of the terms of the polynomial

=> GCF = 4x

Step 2:

Divide each term of the polynomial by the GCF

4x

20x

8x = $\frac{8x}{4x}$ = 2

Step 3:

Use the distributive property to write the polynomial as a product of factors.

=> 4x

The GCF of the terms of the polynomial

=> GCF = 4x

Step 2:

Divide each term of the polynomial by the GCF

4x

^{3 }= $\frac{4x^3}{4x}$ = x^{2}20x

^{2}= $\frac{20x^2}{4x}$ = 5x8x = $\frac{8x}{4x}$ = 2

Step 3:

Use the distributive property to write the polynomial as a product of factors.

=> 4x

^{3}+ 20x^{2}+ 8x = 4x(x^{2}+ 5x + 2)**Question 2:**Factor -16a

^{4}- 40a

^{3}- 24a

^{2}

**Solution:**

Step 1:

The GCF of the terms of the polynomial

=> GCF = - 8a

Step 2:

Factor out the GCF

=> -16a

Step 3:

Factor the trinomial

=> 2a

= 2a(a + 1) + 3(a + 1)

= (2a + 3)(a + 1)

=> Factors of -16a

The GCF of the terms of the polynomial

=> GCF = - 8a

^{2}Step 2:

Factor out the GCF

=> -16a

^{4}- 40a^{3}- 24a^{2}= - 8a^{2}(2a^{2}+ 5a + 3)Step 3:

Factor the trinomial

=> 2a

^{2}+ 5a + 3 = 2a^{2}+ 2a + 3a + 3= 2a(a + 1) + 3(a + 1)

= (2a + 3)(a + 1)

=> Factors of -16a

^{4}- 40a^{3}- 24a^{2}= - 8a^{2}(2a + 3)(a + 1).## How to Factor Polynomial

**Factorization is the breaking apart of a polynomial into a product of other smaller polynomials. Factoring is a way to learn how to take something apart. In order to factor a number or a polynomial, firstly, to break it into factors of the original.**

**Steps for Factoring a Polynomial:**

**Step 1:**If there is a common factor, factor out the GCF.

**Step 2:**Identify the number of terms:

**(i)**If polynomial has

**two terms**, convert polynomial into difference of two squares or sum of two cubes or difference of two cubes.

**(ii)**If polynomial has

**three terms**, convert the polynomial into perfect square trinomial, if not so, then use the trial and check method.

**(iii)**If polynomial has

**four or more terms**, try to factor the polynomial by grouping.

### Solved Examples

**Question 1:**Factor 7x

^{3}- 14x

^{2}- 105x

**Solution:**

Given polynomial, 7x

Step 1:

Find GCF of the terms of the polynomial

=>

Step 2:

Factor out the GCF

=> 7x

Step 3:

Factor the trinomial

x

= x(x - 5) + 3(x - 5)

= (x + 3)(x - 5)

So, the factors of given polynomial is 7x (x + 3)(x - 5)

=> Factors of 7x

^{3}- 14x^{2}- 105xStep 1:

Find GCF of the terms of the polynomial

=>

**GCD**of 7x^{3}, 14x^{2}and 105x is**7x**Step 2:

Factor out the GCF

=> 7x

^{3}- 14x^{2}- 105x = 7x(x^{2}- 2x - 15)Step 3:

Factor the trinomial

x

^{2}- 2x - 15 = x^{2}- 5x + 3x - 15= x(x - 5) + 3(x - 5)

= (x + 3)(x - 5)

So, the factors of given polynomial is 7x (x + 3)(x - 5)

=> Factors of 7x

^{3}- 14x^{2}- 105x = 7x(x + 3)(x - 5).**Question 2:**Factor 2x

^{3}- 2x

^{2}- 4x

**Solution:**

Given polynomial, 2x

Step 1:

Find GCF of the terms of the polynomial

=>

Step 2:

Factor out the GCF

=> 2x

Step 3:

Factor the trinomial

x

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

= (x + 2)(x - 2)

=> Factors of 2x

^{3}- 2x^{2}- 4xStep 1:

Find GCF of the terms of the polynomial

=>

**GCD**of 2x^{3}, 2x^{2}and 4x = 2xStep 2:

Factor out the GCF

=> 2x

^{3}- 2x^{2}- 4x = 2x(x^{2}- x - 2)Step 3:

Factor the trinomial

x

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

= (x + 2)(x - 2)

=> Factors of 2x

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