1. 5.4 Special Factoring Techniques

2. Special Factoring Techniques
By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.
Slide 5.4-3

3. Factor a difference of squares.
Objective 1
Factor a difference of squares.
Slide 5.4-4

4. Factor a difference of squares.
The formula for the product of the sum and difference of the same two terms is
Factoring a Difference of Squares
For example,
The following conditions must be true for a binomial to be a difference of squares:
1. Both terms of the binomial must be squares, such as
x2, 9y2, 25, 1, m4.
2. The second terms of the binomials must have different signs (one positive and one negative).
Slide 5.4-5

5. Factoring Differences of Squares
CLASSROOM EXAMPLE 1
Factoring Differences of Squares
Factor each binomial if possible.
Solution:
After any common factor is removed, a sum of squares cannot be factored.
Slide 5.4-6

6. Factoring Differences of Squares
CLASSROOM EXAMPLE 2
Factoring Differences of Squares
Factor each difference of squares.
Solution:
You should always check a factored form by multiplying.
Slide 5.4-7

7. Factoring More Complex Differences of Squares
CLASSROOM EXAMPLE 3
Factoring More Complex Differences of Squares
Factor completely.
Solution:
Factor again when any of the factors is a difference of squares as in the last problem.
Check by multiplying.
Slide 5.4-8

8. Factor a perfect square trinomial.
Objective 2
Factor a perfect square trinomial.
Slide 5.4-9

9. Factor a perfect square trinomial.
The expressions 144, 4x2, and 81m6 are called perfect squares because A perfect square trinomial is a trinomial that is the square of a binomial. A necessary condition for a trinomial to be a perfect square is that two of its terms be perfect squares. Even if two of the terms are perfect squares, the trinomial may not be a perfect square trinomial.
and
Factoring Perfect Square Trinomials
Slide

10. Factoring a Perfect Square Trinomial
CLASSROOM EXAMPLE 4
Factoring a Perfect Square Trinomial
Factor k2 + 20k
Solution:
Check :
Slide

11. Factoring Perfect Square Trinomials
CLASSROOM EXAMPLE 5
Factoring Perfect Square Trinomials
Factor each trinomial.
Solution:
Slide

12. Factoring Perfect Square Trinomials
1. The sign of the second term in the squared binomial is always the same as the sign of the middle term in the trinomial.
2. The first and last terms of a perfect square trinomial must be positive, because they are squares. For example, the polynomial x2 – 2x – 1 cannot be a perfect square, because the last term is negative.
3. Perfect square trinomials can also be factored by using grouping or the FOIL method, although using the method of this section is often easier.
Slide

13. Factor a difference of cubes.
Objective 3
Factor a difference of cubes.
Slide

14. Factor a difference of cubes.
Factoring a Difference of Cubes
same sign
positive
opposite sign
This pattern for factoring a difference of cubes should be memorized.
The polynomial x3 − y3 is not equivalent to (x − y )3,
whereas
Slide

15. Factoring Differences of Cubes
CLASSROOM EXAMPLE 6
Factoring Differences of Cubes
Factor each polynomial.
Solution:
A common error in factoring a difference of cubes, such as
x3 − y3 = (x − y)(x2 + xy + y2), is to try to factor x2 + xy + y2. It is easy to confuse this factor with the perfect square trinomial x2 + 2xy + y2. But because there is no 2, it is unusual to be able to further factor an expression of the form x2 + xy +y2.
Slide

16. Objective 4
Factor a sum of cubes.
Slide

17. Factoring a Sum of Cubes
Factor a sum of cubes.
A sum of squares, such as m2 + 25, cannot be factored by using real numbers, but a sum of cubes can.
Factoring a Sum of Cubes
same sign
positive
opposite sign
Note the similarities in the procedures for factoring a sum of cubes and a difference of cubes.
1. Both are the product of a binomial and a trinomial.
2. The binomial factor is found by remembering the “cube root, same sign, cube root.”
3. The trinomial factor is found by considering the binomial factor and remembering, “square first term, opposite of the product, square last term.”
Slide

18. Methods of factoring discussed in this section.
Slide

19. Factoring Sums of Cubes
CLASSROOM EXAMPLE 7
Factoring Sums of Cubes
Factor each polynomial.
Solution:
Slide