1. Lesson Objectives: I will be able to …
Choose an appropriate method for factoring a
polynomial
Combine methods for factoring a polynomial
Language Objective: I will be able to …
Read, write, and listen about vocabulary, key
concepts, and examples

2. Not Fully Factored Example: 3x(2x + 4)
Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.
Fully Factored Example: 4x2(3x – 2)
Not Fully Factored Example: 3x(2x + 4)
(2 can also be factored from 2x and 4.)

3. Example 1: Determining Whether a Polynomial is Completely Factored
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Tell whether each polynomial is completely factored. If not factor it.
A. 3x2(6x – 4)
3x2(6x – 4)
6x – 4 can be further factored.
6x2(3x – 2)
Factor out 2, the GCF of 6x and – 4.
6x2(3x – 2) is completely factored.
B. (x2 + 1)(x – 5)
(x2 + 1)(x – 5)
Neither x2 +1 nor x – 5 can be factored further.
(x2 + 1)(x – 5) is completely factored.

4. x2 + 4 is a sum of squares, and cannot be factored.
Caution

5. Tell whether the polynomial is completely factored. If not, factor it.
Your Turn 1
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Tell whether the polynomial is completely factored. If not, factor it.
A. 5x2(x – 1)
Neither 5x2 nor x – 1 can be factored further.
5x2(x – 1) is completely factored.
B. (4x + 4)(x + 1)
(4x + 4)(x + 1)
4x + 4 can be further factored.
4(x + 1)(x + 1)
Factor out 4, the GCF of 4x and 4.
4(x + 1)2 is completely factored.

6. To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.
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7. Example 2: Factoring by GCF and Recognizing Patterns
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Factor the polynomial completely. Check your answer.
10x2 + 48x + 32
2(5x2 + 24x + 16)
Factor out the GCF.
2(5x + 4)(x + 4)
Factor remaining trinomial.
Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)
= 10x2 + 40x + 8x + 32
= 10x2 + 48x + 32 

8. Example 3: Factoring by GCF and Recognizing Patterns
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Factor the polynomial completely. Check your answer.
8x6y2 – 18x2y2
Factor out the GCF. 4x4 – 9 is a perfect-square trinomial of the form a2 – b2.
2x2y2(4x4 – 9)
2x2y2(2x2 – 3)(2x2 + 3)
a = 2x, b = 3
Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9)
= 8x6y2 – 18x2y2 

9. Factor the polynomial completely. Check your answer.
Your Turn 3
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Factor the polynomial completely. Check your answer.
4x3 + 16x2 + 16x
4x(x2 + 4x + 4)
Factor out the GCF.
Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2. a = x, b = 2
4x(x + 2)2
Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4)
= 4x(x2 + 4x + 4)
= 4x3 + 16x2 + 16x 

10. If none of the factoring methods work, the polynomial is said to be unfactorable.
For a polynomial of the form ax2 + bx + c, if there are no numbers whose sum is b and whose product is ac, then the polynomial is unfactorable.
In other words, if the diamond problem is unsolvable, then the polynomial is unfactorable.
Helpful Hint

11. Example 4: Factoring by Multiple Methods
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Factor each polynomial completely.
9x2 + 3x – 2
The GCF is 1 and there is no pattern.
( x + )( x + )
a = 9 and c = –2;
Outer + Inner = 3
Factors of 9 Factors of 2 Outer + Inner
1 and 9
1 and –2
1(–2) + 1(9) = 7 
3 and 3
1 and –2
3(–2) + 1(3) = –3  
3 and 3
–1 and 2
3(2) + 3(–1) = 3
(3x – 1)(3x + 2)

12. Example 5: Factoring by Multiple Methods
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Factor the polynomial completely.
12b3 + 48b2 + 48b
The GCF is 12b; (b2 + 4b + 4) is a perfect-square trinomial in the form of
a2 + 2ab + b2.
12b(b2 + 4b + 4)
(b + )(b + )
Factors of 4 Sum
1 and 
2 and 
a = 2 and c = 2
12b(b + 2)(b + 2)
12b(b + 2)2

13. Example 6: Factoring by Multiple Methods
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Factor the polynomial completely.
4y2 + 12y – 72
Factor out the GCF. There is no pattern. b = 3 and c = –18; look for factors of –18 whose sum is 3.
4(y2 + 3y – 18)
(y + )(y + )
Factors of –18 Sum
–1 and 
–2 and  
–3 and
The factors needed are –3 and 6
4(y – 3)(y + 6)

14. Example 7: Factoring by Multiple Methods.
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Factor the polynomial completely.
(x4 – x2)
x2(x2 – 1)
Factor out the GCF.
x2(x + 1)(x – 1)
x2 – 1 is a difference of two squares.

15. Page 23

16.  Your Turn 7 Factor the polynomial completely. 2p5 + 10p4 – 12p3
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Factor the polynomial completely.
2p5 + 10p4 – 12p3
Factor out the GCF. There is no pattern. b = 5 and c = –6; look for factors of –6 whose sum is 5.
2p3(p2 + 5p – 6)
(p + )(p + )
Factors of – 6 Sum 
The factors needed are –1 and 6
– 1 and
2p3(p + 6)(p – 1)

17. Chapter 8 Quick Review Assignment #16
1. Write the prime factorization of 128.
2. Factor out the GCF: 6x5y2 – 15xy3
3. Factor by grouping: 6a3 – 4a2 – 15a + 10
4. Factor each trinomial.
a) x2 – 11x b) 10x2 – 31x + 15
5. Determine whether each polynomial is a perfect square trinomial, difference of two squares, or neither. If it is a special product, factor it. If not, explain why.
a) 4 – 16x b) 9x2 + 6x + 1
6. Factor completely: -18d2 – 3d + 6

18. Chapter 8 Group Quiz next class!
Homework
Assignment #16
Holt 8-6 #19 – 30, 40 – 45
8-6 Practice C Worksheet
Chapter 8 Group Quiz next class!