Factoring Polynomials

To understand how to factor polynomials, it is first helpful to understand something about factoring numbers.  Factoring numbers is process whereby we determine what numbers are multiplied together to form a given quantity.  For example

18 = (6)(3)

18 = (-6)(-3)

18 = (2)(3)(3)

The last example is considered to be completely factored, since it is broken down into it’s prime factors (2 and 3 are prime numbers).  When factoring polynomials it is done in a similar fashion.  We are trying to determine what factors multiplied together equal the given polynomial.

Methods of Factoring

  1. Given a polynomial ax2+bx+c, where a, b, and c are constants it can be factored by finding all possible factors of c, and then determining which pair of factors add up to b (given that a=1).
  2. a2 – b2 = (a+b)(a-b)
  3. a3 – b3 = (a-b)(a2+ab+b2)
  4. a3 + b3 = (a+b)(a2 – ab+b2)
  5. Factoring by grouping, this is useful when a≠1 from method 1

Examples of Factoring

Factor the following polynomial using method 1

1) x2+5x+4

first start by listing all factors of 4:

4 = (1)(4)

4 = (2)(2)

4 = (-1)(-4)

4 = (-2)(-2)

now we need to determine which set of these factors adds up to the value of b, which is 5

1+4 = 5

it appears that the first group of factors satisfies that condition, hence the factored form of the polynomial is:

x2+5x+4 = (x+1)(x+4)

Factor the following polynomial using method 5

2) 3x2+16x+5

in this case a = 3.  So we will use the method of factoring by grouping.  That means we will now find all possible factors of (a)(c) = (3)(5) = 15

So we need to list all possible factors of 15 which add up to b = 16

15 = (1)(15)

15 = (3)(5)

15 = (-1)(-15)

15 = (-3)(-5)

now we need to determine which set of these factors adds up to the value of b, which is 16

1+15 = 16

So it appears our first group of factors satisfies that condition, now we rewrite the polynomial in the following form

3x2+16x+5 = 3x2+1x+15x+5

note that the polynomial are equivalent.  However, the way in which the new polynomial is written allows us to factor it by grouping

3x2+1x+15x+5 = x(3x+1) + 5(3x+1)

notice that this is the sum of two terms x(3x+1) and 5(3x+1), which both have a similar factor of (3x+1).  That means we can factor out the (3x+1) and we are left with,

x(3x+1)+5(3x+1) = (x+5)(3x+1)

whereby

3x2+16x+5 = (x+5)(3x+1)

Factor the polynomial using method 2

3) 9x2-81

first we need to write this in the form a2-b2

9x2-81 = (3x)2 – 32

where a = 3x, and b=3.  Since

a2 – b2 = (a+b)(a-b)

then

(3x)2 – 32 = (3x+3)(3x-3)

and we can see that

9x2-81 = (3x+3)(3x-3)

Factor the polynomial using method 3

4)  27x3– 64y3

first write this in the form a3-b3

27x3– 64y3 = (3x)3 – (4y)3

where a = 3x, and b = 4y.  Since

a3 – b3 = (a-b)(a2+ab+b2)

then

(3x)3 – (4y)3 = (3x-4y)(9x2+12xy+16y2)

and we can see that

27x3– 64y3 = (3x-4y)(9x2+12xy+16y2)

5) Use quadratic formula: what is the quadratic formula

If in an exam the question reads “solve for x to 2 decimal places” This means it’s time to use the equation. But if we solve for x then we can work back and get the factors.

You might find these short videos on Factorising useful:

  1. Introduction to factorising
  2. Highest common factor (HCF)
  3. Factorising by grouping
  4. Factorising quadratic expressions