Factoring a Trinomial with Two VariablesFactor
Here, the coefficient of zin the middle term is so we need to find two
expressions whose product is - 3 b^2 and whose sum is - 2 b.
- 2 b,
z^2 - 2 bz - 3 b^2.
EXAMPLE 6
SECTION 6.2 Factoring Trinomials 371
NOW TRY
EXERCISE 6
Factor .a^2 + 2 ab- 15 b^2
NOW TRY
EXERCISE 7
Factor. 3 y^4 - 27 y^3 + 60 y^2
factors as
CHECK
=z^2 +zb- 3 bz- 3 b^2 FOIL
1 z - 3 b 21 z+b 2
z^2 - 2 bz- 3 b^21 z- 3 b 21 z +b 2.
Identity and commutative properties=z^2 - 2 bz- 3 b^2 ✓ Combine like terms. NOW TRY
=z^2 + 1 bz- 3 bz- 3 b^2
CAUTION When factoring, always look for a common factor first.Remember
to include the common factor as part of the answer. Always check by multiplying.
Complete solution available
on the Video Resources on DVD
6.2 EXERCISES
In Exercises 1–4, list all pairs of integers with the given product. Then find the pair whose
sum is given. See the tables in Examples 1– 4.
1.Product: 48; Sum: 2.Product: 18; Sum: 9
3.Product: Sum: 4.Product: Sum:
5.Concept Check If a trinomial in xis factored as , what must be true of
aand bif the coefficient of the constant term of the trinomial is negative?1 x+a 21 x+b 2- 24; - 5 - 36; - 16
- 19
NOW TRY ANSWERS
6.
- 3 y^21 y- 521 y- 42
1 a+ 5 b 21 a- 3 b 2Factors of Sums of Factors- 3 b, b - 3 b+b=- 2 b
3 b, -b 3 b+ 1 - b 2 = 2 b- 3 b 2
Sum is- 2 b.OBJECTIVE 2 Factor such trinomials after factoring out the greatest com-
mon factor. If a trinomial has a common factor, first factor it out.
Factoring a Trinomial with a Common FactorFactor
Factor out the greatest
common factor,Factor The integers and have a product of 10 and a sum of
Completely factored formCHECK
FOIL; Combine like terms.= 4 x^5 - 28 x^4 + 40 x^3 ✓ Distributive property NOW TRY
= 4 x^31 x^2 - 7 x+ 102
4 x^31 x- 521 x- 22
= 4 x^31 x- 521 x- 22
x^2 - 7 x+10. - 5 - 2 - 7.
4 x^3.= 4 x^31 x^2 - 7 x+ 102
4 x^5 - 28 x^4 + 40 x^3
4 x^5 - 28 x^4 + 40 x^3.
EXAMPLE 7
Include 4 x^3.