The process by which we determine what has to be multiplied to obtain the given value, which we do many times with the numbers is called factoring polynomials. For example, there are different ways to factories 12.

12 = (2) (6)

12 = (3) (4)

12 = (1/2) (24)

12 = (-2) (-6)

A common technique of factoring numbers is to factor the value into positive prime factors. A prime number is the number whose positive factors are only 1 and itself. 2, 3, 5, 7 are all examples of prime numbers.

If we factor a number into positive prime factors, there is only one method of doing it.

One has to determine all the terms that were multiplied to obtain the given polynomial. Then try to factor every terms that you got in the first step and this continues until you cannot factor further. When you can’t perform any more factoring, it is said that the polynomial is factored completely.

Factoring Polynomial Examples

X² – 16 = (x + 4) (x – 4)

This one is completely factored because neither of the two factors towards the right hand side can be factored any more.

Similarly,

x^4 – 16 = (x² + 4) (x² – 4)

Is not completely factored since the second factor can be factored some more. You can note that the first factor is completely factored. Lets look at the complete factorization of this polynomial.

X^4 – 16 = (x² + 4) (x + 2) (x – 2)

The purpose behind this method is to be familiar with many techniques of factoring polynomials.

**Greatest Common Factor**

Finding the greatest common factor is the basic method for factoring polynomials and it simplifies the problem.

**Strategy for finding the greatest common factor GCF**

- Factor each term completely
- Write a product using each factor that is common to all of the terms
- One each of these factors, use an exponent equal to the smallest exponent that appears on that factor in any of the terms.

To apply this method, look at all the terms and identify whether there is any common factor in all the terms. If any, then factor it out of the polynomial. Also, in this case we are using only the distributive law in reverse. The distributive law states that

a (b + c) = ab + ac

Notice that every term has an ‘a’ and so you factor it out by distributive law in reverse in this manner,

ab + ac = a (b + c)

Greatest Common Factor

Finding the greatest common factor is the basic method for factoring polynomials and it simplifies the problem.

To apply this method, look at all the terms and identify whether there is any common factor in all the terms. If any, then factor it out of the polynomial. Also, in this case we are using only the distributive law in reverse. The distributive law states that

a (b + c) = ab + ac

Notice that every term has an ‘a’ and so you factor it out by distributive law in reverse in this manner,

ab + ac = a (b + c)

Now lets see an example.

8x^4 – 4x³ + 10x²

First, you can notice that the common factor is 2 among all the term. Also, note that we can factor an x2 of every term.

Hence,

8x^4 – 4x³ + 10x² = 2x²(4x² – 2x + 5)

You can check your factoring by multiplying back the terms to make sure that you get the original polynomial.

**Practise This Question**