In some cases, the method of factoring by grouping can be combined with the methods of spe-
cial factoring discussed in this section. Consider this example.Factor within groups.= 12 x+ 3 y 214 x^2 - 6 xy+ 9 y^2 + 2 x- 3 y 2 Combine terms.= 12 x+ 3 y 2314 x^2 - 6 xy+ 9 y^22 + 12 x- 3 y 24= 12 x+ 3 y 214 x^2 - 6 xy+ 9 y^22 + 12 x+ 3 y 212 x- 3 y 2= 18 x^3 + 27 y^32 + 14 x^2 - 9 y^228 x^3 + 4 x^2 + 27 y^3 - 9 y^2SECTION 6.4 A General Approach to Factoring 339
Associative and
commutative propertiesFactor out the greatest
common factor, 2 x+ 3 y.In problems such as this, how we choose to group in the first step is essential to factoring
correctly. If we reach a “dead end,” then we should group differently and try again.Use the method just described to factor each polynomial.
- 64 m^2 - 512 m^3 - 81 n^2 + 729 n^3 78. 10 x^2 + 5 x^3 - 10 y^2 + 5 y^3
8 t^4 - 24 t^3 +t- 3 y^4 +y^3 +y+ 127 a^3 + 15 a- 64 b^3 - 20 b 1000 k^3 + 20 k-m^3 - 2 m125 p^3 + 25 p^2 + 8 q^3 - 4 q 2 27 x^3 + 9 x^2 +y^3 - y^2Factor completely. See Sections 6.1 and 6.2.
- 81.p^2 + 4 p- 21 82. 6 t^2 + 19 ts- 7 s 2
2 ax+ay- 2 bx-by y^2 - y- 2PREVIEW EXERCISES
OBJECTIVES
A General Approach to Factoring
6.4
1 Factor out any
common factor.
2 Factor binomials.
3 Factor trinomials.
4 Factor polynomials
of more than three
terms.
A polynomial is completely factored when bothconditions are satisfied.
1.It is written as a productof prime polynomials with integer coefficients.
2. None of the polynomial factors can be factored further.
Factoring a Polynomial
Step 1 Factor out any common factor.
Step 2 If the polynomial is a binomial,check to see if it is a difference of
squares, a difference of cubes, or a sum of cubes.
If the polynomial is a trinomial,check to see if it is a perfect
square trinomial. If it is not, factor as in Section 6.2.
If the polynomial has more than three terms,try to factor by
grouping.
Step 3 Check the factored form by multiplying.