1. FACTORING
POLYNOMIALS

2. Objective
The student will be able to:
factor perfect square
trinomials.
Designed by Skip Tyler, Varina High School

3. Review: Part I
Multiply.
1. (x + 7)2 = x2 + 14x + 49
2. (x – 2) 2= x2 – 4x + 4
3. (5x + 2y)2= 25x2 + 20xy + 4y2
4. (2x – 9y) 2= 4x2 – 36xy + 81y2
5. (4x + 5y)(4x – 5y)= 16x2 – 25y2
6. (m + 2n)(m – 2n)=
2 2 m4 – 4n2

4. Determine the pattern
1 = 12 These are perfect squares!
4 = 22 You should be able to list
9 = 32 the first 15 perfect
squares in 30 seconds…
16 = 42
25 = 52
Perfect squares
36 = 62 1, 4, 9, 16, 25, 36, 49, 64, 81,
… 100, 121, 144, 169, 196, 225

5. Find the factors
• Factors
x2 + 14x + 49 =(x + 7)2=(x + 7) (x + 7)
x2 – 4x + 4 =(x – 2)2=(x – 2)(x – 2)
25x2 + 20xy + 4y2 =(5x + 2y)(5x + 2y)
4x2 – 36xy + 81y2=(2x – 9y) (2x – 9y)
16x2 – 25y2 =(4x + 5y)(4x – 5y)
m4 – 4n2 =(m2 + 2n)(m2 – 2n)

6. Factoring Chart
This chart will help you to determine
which method of factoring to use.
Type Number of Terms
1. GCF 2 or more
2. Diff. Of Squares 2
3. Trinomials 3

7. Review: Multiply (y + 2)2
Do you remember these?
(y + 2)(y + 2) (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 – 2ab + b2
First terms: y2
Using the formula,
Outer terms: +2y (y + 2)2 = (y)2 + 2(y)(2) + (2)2
Inner terms: +2y (y + 2)2 = y2 + 4y + 4
Last terms: +4 Which one is quicker?
Combine like terms.
y2 + 4y + 4

8. 1) Factor x2 + 6x + 9 Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2
Does this fit the form of our (a - b)2 = a2 – 2ab + b2
perfect square trinomial?
2) Is the first term a perfect
square?
Yes, a = x
2) Is the last term a perfect Since all three are true,
square? write your answer!
Yes, b = 3 (x + 3)2
6) Is the middle term twice the
product of the a and b?
You can still
Yes, 2ab = 2(x)(3) = 6x factor the other way
but this is quicker!

9. 2) Factor y2 – 16y + 64 Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2
Does this fit the form of our (a - b)2 = a2 – 2ab + b2
perfect square trinomial?
2) Is the first term a perfect
square?
Yes, a = y
2) Is the last term a perfect Since all three are true,
square? write your answer!
Yes, b = 8 (y – 8)2
6) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(y)(8) = 16y

10. 3) Factor 4p2 + 4p + 1 =(2p + 1)2
Does this fit the form of our Perfect Square Trinomials
perfect square trinomial? (a + b)2 = a2 + 2ab + b2
2) Is the first term a perfect (a - b)2 = a2 – 2ab + b2
square?
Yes, a = 2p
2) Is the last term a perfectSince all three are true,
square? write your answer!
Yes, b = 1 (2p + 1)2
6) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(2p)(1) = 4p

11. 4) Factor 25x2 – 110xy + 121y2
Does this fit the form of our perfect
square trinomial? Perfect Square Trinomials
(a + b)2 = a2 + 2ab + b2
2) Is the first term a perfect (a - b)2 = a2 – 2ab + b2
square?
Yes, a = 5x
4) Is the last term a perfect Since all three are true,
square? write your answer!
Yes, b = 11y (5x – 11y)2
6) Is the middle term twice the
product of the a and b?
Yes, 2ab = 2(5x)(11y) = 110xy

12. FACTORING
POLYNOMIALS
Factor the following: Factor the following:
•g2 – 24g + 144= =(g-12 ) (g-12 )
•m2 + 10m + 25 = (m+5)(m+5)
•4a2 – 24a+ 36 = (2m-6)(2a-6)

13. Factor m – 12m + 36
2
• (m – 6)(m + 6)
• (m – 6)2
• (m + 6)2
• (m – 18)2

14. Factor 2r + 12r + 18
2
• prime
• 2(r2 + 6r + 9)
• 2(r – 3)2
• 2(r + 3)2
• 2(r – 3)(r + 3)
Don’t forget to factor the
GCF first!

15. Review
Factor 9k2 + 12k + 4
• (3k + 2)2 = (3k + 2) (3k + 2)
• (3k – 2)2 = (3k – 2) (3k – 2)
• (3k + 2)(3k – 2) = (3k + 2)(3k – 2)
• I’ve got no clue…I’m lost!

16. Find the factors:
= (x + 3) (x + 3)
• x² + 6x + 9 = (x + 3)²
• 9m² - 6m + 1 = (3m - 1)² = (3m - 1) (3m - 1)
• 25s² - 20s + 4 = (5s - 2)² = (5s - 2) (5s - 2)
• x² + 2xy + y² = (x + y)² = (x + y) (x + y)

17. Find the complete factors:
1. 4x² + 4x + 1 = = (x + 3) (x + 3)
3. 25s² - 60s + 36 = = (3m - 1) (3m - 1)
5. 27m² - 18m +3 =
= (5s - 2) (5s - 2)
=
8. 8x² -72xy + 162y² = (x + y) (x + y)