Section 4.4 Factoring Trinomials
RECALL multiplying using the FOIL method:
(3𝑥 − 8)(2𝑥 + 3)
REVERSE FOIL:
1) List all pairs of factors that multiply together to give the FIRST term. These will be the first terms in
each binomial. (F
)(F
).
*Not sure which pair to try first? Start the pair that is closest together. If that pair doesn’t work, then we will try another pair.
2) Put the SIGNS in the middle of each binomial. Look at the sign of the LAST term.
Is it “+?” The signs will be the same. They will both be the sign of the middle term.
𝐴𝑥 + 𝐵𝑥 + 𝐶 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑠 ( + )( + )
𝐴𝑥 − 𝐵𝑥 + 𝐶 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑠 ( − )( − )
Is it “−?” The signs will be opposites.
𝐴𝑥 − 𝐵𝑥 − 𝐶
OR
𝐴𝑥 + 𝐵𝑥 − 𝐶
𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑎𝑠 ( − )( + ) or ( − )( + )
3) List all pairs of factors that multiply together to give the LAST term. These will be the last terms in
each binomial. (
L )(
L)
*Not sure which pair to try first? Start the pair that is closest together. If that pair doesn’t work, then we will try another pair.
**NOTE: There cannot be common factors within a single binomial!!
4) Determine if the O and the I combine to give the original middle term.
*Right number, but wrong sign? Switch the signs.
Factor using REVERSE FOIL.
6𝑥 − 7𝑥 − 24
Example 1: Completely factor the following trinomials.
a) −2𝑥 + 20𝑥 − 18
b) 75𝑥 − 90𝑥 𝑦 + 27𝑥𝑦
c) 𝑥 + 7𝑥 + 4
d) 9𝑎 + 62𝑎𝑏 − 7𝑏
e) 6𝑥 (2𝑥 + 3) + 𝑥𝑏(2𝑥 + 3) − 2𝑏 (2𝑥 + 3) f) 15𝑥 (𝑥 + 1) − 33𝑥 (𝑥 + 1) + 6𝑥(𝑥 + 1)
g) 6𝑥 + 𝑥 − 12
h) 2 − 4𝑟 − 30𝑟