NOTEThe factored form in Example 8can be written in other ways, such as
and
Verify that these both give the original trinomial when multiplied.
Factoring a Trinomial with a Common FactorFactor
GCF 8 y= 8 y 12 y- 121 y+ 22 Factor the trinomial.
= 8 y 12 y^2 + 3 y- 22 =
16 y^3 + 24 y^2 - 16 y
16 y^3 + 24 y^2 - 16 y.
EXAMPLE 9
1 - 3 x- 221 x- 62 13 x+ 221 - x+ 62.
SECTION 6.2 Factoring Trinomials 331
NOW TRYOBJECTIVE 4 Factor by substitution.
Factoring a Polynomial by SubstitutionFactor
Since the binomial appears to powers 2 and 1, we let a substitution variable
represent We may choose any letter we wish except x. We choose t.
Let
Factor.
Replace twith
Simplify.= 12 x+ 321 x+ 72 NOW TRY
= 12 x+ 6 - 321 x+ 72
= 321 x+ 32 - 3431 x+ 32 + 44 x+3.
= 12 t- 321 t+ 42
= 2 t^2 + 5 t- 12 t=x+3.
21 x+ 322 + 51 x+ 32 - 12
x+3.
x+ 3
21 x+ 322 + 51 x+ 32 - 12.
EXAMPLE 10
CAUTION Remember to make the final substitution of for t in
Example 10.
x+ 3
NOW TRY
EXERCISE 9
Factor 12y^3 + 33 y^2 - 9 y.
NOW TRY ANSWERS
- 3 y 14 y- 121 y+ 32
NOW TRY
EXERCISE 10
Factor.
31 a+ 222 - 111 a+ 22 - 4- 13 a+ 721 a- 22
Factoring a Trinomial in FormFactor
The variable yappears to powers in which the greater exponent is twice the lesser
exponent. We can let a substitution variable represent the variable to the lesser power.
Here, we let
Substitute tfor.
Factor.= 13 y^2 - 4212 y^2 + 52 t=y 2
= 13 t- 4212 t+ 52
= 6 t^2 + 7 t- 20 y^2
= 61 y^222 + 7 y^2 - 20 y^4 = 1 y^222
6 y^4 + 7 y^2 - 20
t=y^2.
6 y^4 + 7 y^2 - 20.
EXAMPLE 11 ax (^4) +bx (^2) +c
Don’t stop here.
Replace twith y^2.
NOW TRY
NOTE Some students feel comfortable factoring polynomials like the one in
Example 11directly, without using the substitution method.
NOW TRY
EXERCISE 11
Factor 6x^4 + 11 x^2 +3.
- 13 x^2 + 1212 x^2 + 32
Remember the
common factor.