2. Chapter 7
Factoring

3. 7.3
Special Factoring

4. 7.3 Special Factoring Objectives
Factor a difference of squares.
Factor a perfect square trinomial.
Factor a difference of cubes.
Factor a sum of cubes.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

5. The Difference of Squares
7.3 Special Factoring
The Difference of Squares
Difference of Squares
x2 – y2 = (x + y)(x – y)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

6. Factoring Differences of Squares
7.3 Special Factoring
EXAMPLE 1
Factoring Differences of Squares
Factor each polynomial.
(a) 2n2 – 50
There is a common factor of 2.
2n2 – = 2(n2 – 25) Factor out the common factor.
= 2(n + 5)(n – 5) Factor the difference of squares.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

7. Factoring Differences of Squares
7.3 Special Factoring
EXAMPLE 1
Factoring Differences of Squares
Factor each polynomial.
(b) 9g2 – 16
9g2 – 16
= (3g)2 – (4)2
= (3g + 4)(3g – 4) A2 – B2
(A B)(A – B)
(c) 4h2 – (w + 5)2
4h2 – (w + 5)2
= (2h)2 – (w + 5)2
= (2h + w + 5)(2h – [w + 5])
= (2h + w + 5)(2h – w – 5) A2 B2 –
(A B) (A – B)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

8. 7.3 Special Factoring Caution
Assuming no greatest common factor except 1, it is not possible to
factor (with real numbers) a sum of squares, such as x
Copyright © 2010 Pearson Education, Inc. All rights reserved.

9. Perfect Square Trinomial
7.3 Special Factoring
Perfect Square Trinomial
Perfect Square Trinomial
x xy + y2 = (x + y)2
x2 – 2xy + y2 = (x – y)2
Copyright © 2010 Pearson Education, Inc. All rights reserved.

10. Factoring Perfect Square Trinomials
7.3 Special Factoring
EXAMPLE 2
Factoring Perfect Square Trinomials
Factor each polynomial.
(a) 9g2 – 42g
Here 9g2 = (3g)2 and 49 = 72. The sign of the middle term is –, so if
9g2 – 42g is a perfect square trinomial, the factored form will
have to be
(3g – 7)2.
Take twice the product of the two terms to see if this is correct.
2(3g)(–7) = –42g
This is the middle term of the given trinomial, so
9g2 – 42g = (3g – 7)2.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

11. Factoring Perfect Square Trinomials
7.3 Special Factoring
EXAMPLE 2
Factoring Perfect Square Trinomials
Factor each polynomial.
(b) 25x xy y2
If this is a perfect square trinomial, it will equal (5x + 8y)2. By the
pattern described earlier, if multiplied out, this squared binomial has a
middle term of 2(5x)(8y), which does not equal 60xy. Verify that this
trinomial cannot be factored by the methods of the previous section
either. It is prime.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

12. Factoring Perfect Square Trinomials
7.3 Special Factoring
EXAMPLE 2
Factoring Perfect Square Trinomials
Factor each polynomial.
(c) (n – 4) (n – 4)
= [ (n – 4) ]2
= (n + 5)2,
since 2(n – 4)9 = 18(n – 4), the middle term.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

13. Factoring Perfect Square Trinomials
7.3 Special Factoring
EXAMPLE 2
Factoring Perfect Square Trinomials
Factor each polynomial.
(d) c2 – 6c – h2
Since there are four terms, we will use factoring by grouping. The first
three terms here form a perfect square trinomial. Group them together,
and factor as follows.
(c2 – 6c + 9) – h2
= (c – 3)2 – h2
The result is the difference of squares. Factor again to get
= (c – h)(c – 3 – h).
Copyright © 2010 Pearson Education, Inc. All rights reserved.

14. 7.3 Special Factoring Difference of Cubes
x3 – y3 = (x – y)(x2 + xy + y2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

15. Factoring Difference of Cubes
7.3 Special Factoring
EXAMPLE 3
Factoring Difference of Cubes
Factor each polynomial. Recall, x3 – y3 = (x – y)(x2 + xy + y2).
(a) a3 – 125
= a3 – 53
= (a – 5)(a a )
= (a – 5)(a a ) a3
–125
Check:
= (a – 5)(a a )
–5a
Opposite of the product of the cube
roots gives the middle term.
Copyright © 2010 Pearson Education, Inc. All rights reserved.

16. Factoring Difference of Cubes
7.3 Special Factoring
EXAMPLE 3
Factoring Difference of Cubes
Factor each polynomial. Recall, x3 – y3 = (x – y)(x2 + xy + y2).
(b) 8g3 – h3
= (2g)3 – h3
= (2g – h) [ (2g)2 + (2g)(h) + h2) ]
= (2g – h)(4g gh + h2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

17. Factoring Difference of Cubes
7.3 Special Factoring
EXAMPLE 3
Factoring Difference of Cubes
Factor each polynomial. Recall, x3 – y3 = (x – y)(x2 + xy + y2).
(c) 64m3 – 27n3
= (4m)3 – (3n)3
= (4m – 3n) [ (4m)2 + (4m)(3n) + (3n)2 ]
= (4m – 3n)(16m mn + 9n2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

18. 7.3 Special Factoring Sum of Cubes
x3 + y3 = (x + y)(x2 – xy + y2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

19. 7.3 Special Factoring Note on Signs
The sign of the second term in the binomial factor of a sum or difference
of cubes is always the same as the sign in the original polynomial.
In the trinomial factor, the first and last terms are always positive;
the sign of the middle term is the opposite of the sign of the second term
in the binomial factor.
Difference of Cubes
x3 – y3 = (x – y)(x2 + xy + y2)
Sum of Cubes
x3 + y3 = (x + y)(x2 – xy + y2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

20. Factoring Sums of Cubes
7.3 Special Factoring
EXAMPLE 4
Factoring Sums of Cubes
Factor each polynomial. Recall, x3 + y3 = (x + y)(x2 – xy + y2).
(a) n
= n
= (n + 2)(n2 – 2n )
= (n + 2)(n2 – 2n + 4)
(b) 64v g3
= (4v)3 + (3g)3
= (4v + 3g) [ (4v)2 – (4v)(3g) + (3g)2 ]
= (4v + 3g) (16v2 – 12gv + 9g2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.

21. Factoring Sums of Cubes
7.3 Special Factoring
EXAMPLE 4
Factoring Sums of Cubes
Factor each polynomial. Recall, x3 + y3 = (x + y)(x2 – xy + y2).
(c) 2k =
= 2(k )
= 2(k )
= 2(k + 5)(k2 – 5k )
Copyright © 2010 Pearson Education, Inc. All rights reserved.

22. 7.3 Special Factoring Factoring Summary
Special Types of Factoring (Memorize)
Difference of Squares x2 – y2 = (x + y)(x – y)
Perfect Square Trinomial x xy + y2 = (x + y)2
x2 – 2xy + y2 = (x – y)2
Difference of Cubes x3 – y3 = (x – y)(x2 + xy + y2)
Sum of Cubes x3 + y3 = (x + y)(x2 – xy + y2)
Copyright © 2010 Pearson Education, Inc. All rights reserved.