FACTORING
GUIDELINES
Pre-requirement for Chapter 1
3 Guidelines you will need…
■ Greatest Common Factor
■ Two Terms
■ Three Terms
Greatest Common Factor
■ Always check for the Greatest Common Factor first.
■ Are there any common factors? If so, factor out the
common factor.
3𝑦 2 + 9𝑦 = 3𝑦 𝑦 + 3
8𝑎 3 − 4𝑎 2 = 4𝑎 2 (2𝑎 − 1)
6𝑥 2 + 3𝑥 + 12 = 3(2𝑥 2 + 𝑥 + 4)
Special Two Term Factoring
■ If there are two terms, decide if one of the following could be applied.
– Difference of two squares: If in this form 𝑎2 − 𝑏2 then
(𝑎 − 𝑏(𝑎 + 𝑏) will be the form of the answer.
– Difference of two cubes: If in this form 𝑎3 − 𝑏3 then
(𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2 ) will be the form of the answer.
– Sum of two cubes: If in this form 𝑎3 + 𝑏3 then
(𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2 ) will be the form of the answer.
Difference of Two Squares
■ 4𝑥 2 − 16 = 4(𝑥 2 − 4)
Factor out the GCF first
= 4(𝑥 − 2)(𝑥 + 2)
After factoring out the GCF, check your
answer to see if it can be factored again.
■ 4𝑥 2 − 49 = (2𝑥 − 7)(2𝑥 + 7)
■ 9𝑏 2 − 16𝑐 2 = (3𝑏 − 4𝑐)(3𝑏 + 4𝑐)
Difference of Two Cubes
𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2 )
■ 8𝑥 3 − 27
First find the cube root of each term. That gives you the a and b
that you need to plug into the form above.
𝑎 = 2𝑥 and 𝑏 = 3
8𝑥 3 − 27 = (2𝑥 − 3)(4𝑥 2 + 6𝑥 + 9)
Factoring Three Terms
2
Factor 𝑥 + 𝑏𝑥 + 𝑐
■ The factored form is:
(𝑥 + 𝑜𝑛𝑒 𝑛𝑢𝑚𝑏𝑒𝑟)(𝑥 + 𝑜𝑡ℎ𝑒𝑟 𝑛𝑢𝑚𝑏𝑒𝑟)
■ Identify the factors that multiply to give the ‘c’ value (the
constant term) but also give the ‘b’ value (the coefficient of x)
2
Example for 𝑥 + 𝑏𝑥 + 𝑐
■ 𝑥 2 + 7𝑥 + 12 = (𝑥+
)(𝑥+
■ The factors of 12 are:
(1, 12), (2, 6), (3, 4)
■ 𝑥 2 + 7𝑥 + 12 = (𝑥 + 3)(𝑥 + 4)
■ Try 𝑥 2 + 7𝑥 + 10
)
2
Factor 𝑎𝑥 + 𝑏𝑥 + 𝑐
■ Multiply ‘a’ with ‘c’ and call this value as 𝑎𝑐.
■ Write the factors of 𝑎𝑐 that will also add to give the 𝑏 value.
■ Write these values as the coefficients of separate 𝑥 terms.
■ ….
■ Let’s work the rest of steps with an example.
2
Example for 𝑎𝑥 + 𝑏𝑥 + 𝑐
■ 3𝑥 2 + 7𝑥 + 2
■ Multiply 3 2 = 6
■ List of all the factors of 6
(1, 6), (2, 3)
3𝑥 2 + 7𝑥 + 2
= 3𝑥 2 + 𝑥 + 6𝑥 + 2
= 𝑥 3𝑥 + 1 + 2 3𝑥 + 1
= (3𝑥 + 1)(𝑥 + 2)
Try
1) 2𝑥 + 6
2) 5 − 10𝑧
3) −8𝑝 + 16𝑝2
4) −6𝑒 3 − 18𝑒 2
5) −7𝑧 + 21𝑧 2
6) 𝑧 2 − 1
7) 4𝑝2 − 25𝑞 2
8) 16𝑠 2 − 49𝑡 2
9) 5𝑥 2 − 125𝑦 2
10) 2𝑝2 − 200𝑞 2
11) 𝑥 2 + 4𝑥 + 4
12) 𝑥 2 + 3𝑥 + 2
13) 𝑥 2 − 10𝑥 + 21
14) −5𝑥 + 𝑥 2 − 24
15) 𝑥 2 + 30 − 11𝑥
16) 4𝑥 2 + 16𝑥 + 12
17) 2𝑥 2 + 3𝑥 + 1
18) 4𝑥 2 − 19𝑥 + 12