The methods of factoring discussed in this section are summarized here.
SECTION 6.4 Special Factoring Techniques 387
Special FactorizationsDifference of squares
Perfect square trinomials
Difference of cubes
Sum of cubes
The sum of squares can be factored only if the terms have a common factor.
x^3 y^3 1 xy 21 x^2 xyy^22
x^3 y^3 1 xy 21 x^2 xyy^22
x^2 2 xyy^2 1 xy 22
x^2 2 xyy^2 1 xy 22
x^2 y^2 1 xy 21 xy 2
Complete solution available
on the Video Resources on DVD
6.4 EXERCISES
1.Concept Check To help you factor the difference of squares, complete the following list
of squares.2.Concept Check The following powers of xare all perfect squares: , , , ,.
On the basis of this observation, we may make a conjecture (an educated guess) that if
the power of a variable is divisible by (with 0 remainder), then we have a perfect
square.
3.Concept Check To help you factor the sum or difference of cubes, complete the fol-
lowing list of cubes.4.Concept Check The following powers of xare all perfect cubes: , , , ,. On
the basis of this observation, we may make a conjecture that if the power of a variable is
divisible by (with 0 remainder), then we have a perfect cube.
5.Concept Check Identify each monomial as a perfect square, a perfect cube, both of
these, or neither of these.
(a) (b) (c) (d)
6.Concept Check What must be true for to be both a perfect square and a perfect cube?Factor each binomial completely. If the binomial is prime, say so. Use your answers from
Exercises 1 and 2as necessary. See Examples 1–3.
- 4 x^2 - 9 17. 36 x^2 - 16 18. 32 a^2 - 8
4 m^2 + 16 9 x^2 + 81 9 r^2 - 4x^2 - 400 m^2 + 64 k^2 + 49y^2 - 25 t^2 - 16 x^2 - 144xn64 x^6 y^12125 t^649 x^1281 r^10x^3 x^6 x^9 x^12 x^1563 = 73 = 83 = 93 = 103 =
13 = 23 = 33 = 43 = 53 =
x^2 x^4 x^6 x^8 x^10