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)