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How to Factor a Polynomial

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

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To factor a polynomial means to write the polynomial as a product of other polynomials. The distributive property is used to multiply factors of a polynomial. The distributive property is used to factor a common binomial factor from an expression. The process of writing a polynomial as a product of several factors is called as factoring. If a polynomial cannot be factored by using integers only, such type of polynomials are called prime polynomials. While factorizing polynomials expressions, firstly find the GCF. The greatest common factor of two or more monomials is the product of the coefficients and the common variable factors. The exponent of each variable in the GCF is the same as the smallest exponent of that variable in any of the monomials.

Solved Examples

Question 1: Find the factors of 4x3 + 20x2 + 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= $\frac{4x^3}{4x}$ = x2

20x2 = $\frac{20x^2}{4x}$ = 5x

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

Step 3:
Use the distributive property to write the polynomial as a product of factors.

=> 4x3 + 20x2 + 8x = 4x(x2 + 5x + 2)
 

Question 2: Factor -16a4 - 40a3 - 24a2 
Solution:
Step 1:

The GCF of  the terms of the polynomial

=> GCF = - 8a2

Step 2:
Factor out the GCF

=> -16a4 - 40a3 - 24a2 = - 8a2 (2a2 + 5a + 3)

Step 3:
Factor the trinomial

=> 2a2 + 5a + 3 = 2a2 + 2a + 3a + 3

= 2a(a + 1) + 3(a + 1)

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

=> Factors of -16a4 - 40a3 - 24a2 = - 8a2 (2a + 3)(a + 1).
 

How to Factor Polynomial

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A polynomial is factored when it is  written as a product of non factorable factors. The first step in factoring any polynomial is to determine whether the terms of the polynomial have a greatest common factor. If yes, factor it out first. 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 7x3 - 14x2 - 105x
Solution:
Given polynomial,  7x3 - 14x2 - 105x

Step 1:
Find GCF of the terms of the polynomial

=> GCD of 7x3 , 14x2 and 105x is 7x

Step 2:
Factor out the GCF

=> 7x3 - 14x2 - 105x = 7x(x2 - 2x - 15)

Step 3:
Factor the trinomial

x2 - 2x - 15 = x2 - 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 7x3 - 14x2 - 105x = 7x(x + 3)(x - 5).
 

Question 2: Factor 2x3 - 2x2 - 4x
Solution:
Given polynomial,  2x3 - 2x2 - 4x

Step 1:
Find GCF of the terms of the polynomial

=> GCD of  2x3 , 2x2 and 4x = 2x

Step 2:
Factor out the GCF

=> 2x3 - 2x2 - 4x = 2x(x2 - x - 2)

Step 3:
Factor the trinomial

x2 - x - 2 = x2 - 2x + x - 2

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

= (x + 2)(x - 2)

=> Factors of 2x3 - 2x2 - 4x = 2x (x + 2)(x - 2)