5.4 Special Factoring Techniques

By reversing the rules for multiplication of binomials from Section 4.6, we get rules for factoring polynomials in certain forms.

Special Factoring Techniques

Slide 5.4-3

Objective 1

Factor a difference of squares.

Slide 5.4-4

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

2 2.x y x y x y

2 2x y x y x y

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).

2 2 216 4 4 4 .m m m m For example,

Slide 5.4-5

Factor each binomial if possible.

Solution:

2 81t

2 2r s2 10y 2 36q

9 9t t

r s r s

prime

prime

After any common factor is removed, a sum of squares cannot be factored.

Slide 5.4-6

Factoring Differences of SquaresCLASSROOM EXAMPLE 1

Factor each difference of squares.

Solution:

249 25x

2 264 81a b

7 5 7 5x x

8 9 8 9a b a b

You should always check a factored form by multiplying.

Slide 5.4-7

Factoring Differences of SquaresCLASSROOM EXAMPLE 2

Factor completely.

2 9 3 3z z z

Solution:

250 32r

4 100z

4 81z

22 25 16r

2 210 10z z

2 29 9z z

2 5 4 5 4r r

Factor again when any of the factors is a difference of squares as in the last problem.

Check by multiplying.

Slide 5.4-8

Factoring More Complex Differences of SquaresCLASSROOM EXAMPLE 3

Objective 2

Factor a perfect square trinomial.

Slide 5.4-9

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.

Factoring Perfect Square Trinomials

22 22x xy y x y

22 22x xy y x y

2144 ,12 224 2 ,x x 26 381 9 .m mand

Slide 5.4-10

Solution:

210k

Factor k2 + 20k + 100.

2 20 100k k

Check :

2 10 20k k

Slide 5.4-11

Factoring a Perfect Square TrinomialCLASSROOM EXAMPLE 4

Factor each trinomial.

Solution:2 24 144x x

225 30 9x x

236 20 25a a

212x

25 3x

prime

3 218 84 98x x x 22 9 42 49x x x 22 3 7x x

Slide 5.4-12

Factoring Perfect Square TrinomialsCLASSROOM EXAMPLE 5

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.

Factoring Perfect Square Trinomials

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.

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.

Slide 5.4-13

Objective 3

Factor a difference of cubes.

Slide 5.4-14

Factor a difference of cubes.

The polynomial x3 − y3 is not equivalent to (x − y )3,

whereas

3x y x y x y x y

2 22x y x xy y

3 3 2 2x y x y x xy y

Factoring a Difference of Cubes

3 3 2 2x y x y x xy y same sign

positive

opposite sign

Slide 5.4-15

This pattern for factoring a difference of cubes should be memorized.

Factor each polynomial.

Solution:

3 216x

327 8x 35 5x

26 6 36x x x

23 2 9 6 4x x x

3 664 125x y 2 2 2 44 5 16 20 25 x y x xy y

35 1x 25 1 1x x x

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 5.4-16

Factoring Differences of CubesCLASSROOM EXAMPLE 6

Objective 4

Factor a sum of cubes.

Slide 5.4-17

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.

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.”

Factoring a Sum of Cubes

3 3 2 2x y x y x xy y

same sign

positive

opposite sign

Slide 5.4-18

Methods of factoring discussed in this section.

Slide 5.4-19

Factor each polynomial.

Solution:

3 64p

3 327 64x y

6 3512a b

24 4 16p p p

2 23 4 9 12 16x y x xy y

2 4 2 28 64 8a b a a b b

Slide 5.4-20

Factoring Sums of CubesCLASSROOM EXAMPLE 7