Intermediate Algebra (11th edition)

(b)

If this is a perfect square trinomial, it will equal By the pattern in

the box, if multiplied out, this squared binomial has a middle term of

which does not equal 20 mn. Verify that this trinomial cannot be factored by the meth-

ods of the previous section either. It is prime.

(c)

the middle term.

(d)

Since there are four terms, use factoring by grouping. The first three terms here

form a perfect square trinomial. Group them together, and factor as follows.

Factor the perfect square trinomial.

Factor the difference of squares.

NOW TRY

= 1 m- 4 + p 21 m- 4 - p 2

= 1 m- 422 - p^2

= 1 m^2 - 8 m+ 162 - p^2

m^2 - 8 m+ 16 - p^2

m^2 - 8 m+ 16 - p^2

= 1 r + 822

= 31 r+ 52 + 342 21 r+ 52132 = 61 r+ 52 ,

1 r+ 522 + 61 r + 52 + 9

212 m 217 n 2 = 28 mn,

12 m+ 7 n 22.

4 m^2 + 20 mn + 49 n^2

SECTION 6.3 Special Factoring 335

NOW TRY
EXERCISE 2
Factor each polynomial.

(a)

(b)

(c)y^2 - 16 y+ 64 - z^2

16 x^2 - 56 xy+ 49 y^2

a^2 + 12 a+ 36

NOW TRY ANSWERS

2. (a)
   (b)
   (c) 1 y- 8 +z 21 y- 8 - z 2

14 x- 7 y 22

1 a+ 622

Difference of Cubes

x^3 y^3  1 xy 21 x^2 xyy^22

Check by showing that the product of x- yand x^2 + xy+ y^2 is x^3 - y^3.

Factoring Differences of Cubes

Factor each polynomial.

(a)

CHECK

Distributive property

=m^3 - 8 ✓ Combine like terms.

=m^3 + 2 m^2 + 4 m- 2 m^2 - 4 m- 8

1 m- 221 m^2 + 2 m+ 42

= 1 m- 221 m^2 + 2 m+ 42

m^3 - 8 = m^3 - 23 = 1 m- 221 m^2 +m# 2 + 222

x^3 - y^3 = 1 x - y 21 x^2 +^ x # y + y^22

EXAMPLE 3

NOTE Perfect square trinomials can be factored by the general methods shown ear-

lier for other trinomials. The patterns given here provide “shortcuts.”

OBJECTIVE 3 Factor a difference of cubes.A difference of cubes,

can be factored as follows.

x^3 - y^3 ,