Factor.

64x144x81 2

8-6 Perfect Squares and Factoring

Algebra 1 Glencoe McGraw-Hill Linda Stamper

In the previous lesson you worked with the difference of two squares (sum and difference pattern). Today the focus is on perfect square trinomials (square of a binomial pattern).

For a trinomial to be factorable as a perfect square, three conditions must be satisfied.

25x20x4 2

The first term must be a

perfect square.

The middle term must be twice the

product of the square roots of the first and last terms.

The last term must be a

perfect square.

2x4 x225 5

x105x2 x202x10

25x2

2x

x +

2Remember the square of a binomial is a SQUARE ?

4x4x2x 22

3x x +

3

9x6x3x 22

4x x +

4

16x8x4x 22

To find the product…

16x8x4x

9x6x3x

4x4x2x

22

22

22

25x

Square first term.

Multiply the terms and then

double it.

Square last term.

25x10x2 26x 36x12x2

22x2 4x8x4 2 23x2 9x12x4 2 25x3 25x30x9 2 26x4 36x48x16 2

Factoring is working backwards from multiplying!

Think – What is the square of the

first term and what is the

square of the second term.

22x2 Can you

factor out a greatest monomial

factor?

Square of a Binomial Pattern

16x8x4x

9x6x3x

4x4x2x

22

22

22

25x 25x10x2

26x 36x12x2

4x8x4 2 23x2 9x12x4 2

25x3 25x30x9 2

26x4 36x48x16 2

Which answer/s are not factored

completely?

21x4

1x2 1x22x2 2x2

23x24

3x22 3x226x4 6x4

21x4

1x2 1x22x2 2x2

Could you use the X-Factor method to

factor these problems?

Factor.

64x144x81 2 28x9

8x98x9

Do you recognize this Square of a Binomial Pattern?

16x8x4x

9x6x3x

4x4x2x

22

22

22

Square first term.

Multiply the terms and then double it. Note: middle

term is negative!

Square last term.

2x x –

2The middle term is always negative?

4x4x2x 22

Square of a Binomial Pattern

16x8x4x

9x6x3x

4x4x2x

22

22

22

25x 25x10x2 26x 36x12x2

22x2 4x8x4 2 24x3 16x24x9 2

25x4 25x40x16 2

Which answer/s are not factored

completely?

21x4

1x2 1x2

2x2 2x2

Example 1 100y60y9 2

210y3

Check using FOIL!

Determine whether each trinomial is a perfect square trinomial. If so, factor it.

Example 2 81m144m64 2 29m8

Example 3

4m12m9 2 22m3

Example 4 64y32y16 2

.T.S.Panot

Example 5 25y40y16 2 25y4

Example 6 x6036x25 2

26x5

Example 7 9n24n16 2

23n4

Example 8 81y9y2

.T.S.Panot

Example 9 9x30x25 2

23x5

When solving equations involving repeated factors, it is only necessary to set one of the repeated factors equal to zero.

2516

x58

x2

02516

x58

x2

054

x2

054

x

54

x

or054

x

58

54

2

Double

root!

Use factoring to solve the equation. Remember if it is a perfect square, it is only necessary to set one of the repeated factors equal to zero

036a12a2

Example 10 Example 11

094

y34

y2

x3681x4 2

Example 12 Example 13 16d24d9 2

x9x12x4 23

Example 14 Example 15

41

xx2

036a12a2

Example 10 Example 11

094

y34

y2

6a

32

y

06a 2 06a 0

32

y2

032

y

x3681x4 2

Example 12 Example 13 16d24d9 2

29

x

34

d

081x36x4 2

09x2 2 09x2

9x2

016d24d9 2

04d3 2 04d3

4d3

x9x12x4 23

Example 14 Example 14

23

x

0x9x12x4 3

09x12x4x 2

03x2x 2

3x2 03x2or0x

41

xx2

041

xx2

021

x2

021

x

21

x

HOW TO APPROACH FACTORING PROBLEMS

When factoring it is important to approach the problem in the following order. Look at the problem carefully and ask yourself:1. Can I pull out a greatest common factor? If you can, do it, but then look at what is left to check for opportunities to factor again.2. Is this the DIFFERENCE of Two Squares?

22 ba Remember that these never have a middle (x) term. If so, factor it into the ( + )( – ) pattern.

3. Is this a Perfect Square Trinomial? 22

22

bab2a

or bab2a

If so, it will factor into two parentheses that are exactly the same and can be written as a square. 22 3x or 3x

Check by doing FOIL. Remember that factoring is working backwards from multiplying.

Always look at what remains and check for opportunities to factor further.

4. If you examined the problem closely and answered the above

questions each with a NO, then you must factor using the X figure. Use the X-Factor method if the leading coefficient does not equal 1.

HOW TO APPROACH FACTORING PROBLEMS

5. If there are four or more terms look to factor by grouping. Group terms with common factors and then factor out the GCF from each grouping. Then use the distributive property a second time to factor a common binomial factor.

8-A12 Pages 458-460 #12–30,57–60.