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
- 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).
- a2 – b2 = (a+b)(a-b)
- a3 – b3 = (a-b)(a2+ab+b2)
- a3 + b3 = (a+b)(a2 – ab+b2)
- 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:
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:
- Introduction to factorising
- Highest common factor (HCF)
- Factorising by grouping
- Factorising quadratic expressions