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Warm Up
Lesson Presentation

2. Warm Up
Factor each trinomial.
1. x2 + 13x + 40
(x + 5)(x + 8)
2. 5x2 – 18x – 8
(5x + 2)(x – 4)
3. Factor the perfect-square trinomial 16x2 + 40x + 25.
(4x + 5)(4x + 5)
4. Factor 9x2 – 25y2 using the difference of
two squares.
(3x + 5y)(3x – 5y)

3. Recall that a polynomial is fully or completely factored when it is written as a product of monomials and polynomials whose terms have no common factors other than 1.

4. Additional Example 1: Determining Whether an Expression is Completely Factored
Tell whether each expression is completely factored. If not, factor it.
A. 3x2(6x – 4)
3x2(6x – 4)
6x – 4 can be factored further.
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.

5. Additional Example 1: Determining Whether an Expression is Completely Factored
Tell whether each expression is completely factored. If not, factor it.
C. 5x(x2 – 2x – 3)
(x2 – 2x – 3) can be factored further.
5x(x2 – 2x – 3)
5x(x + 1)(x – 3)
Factor x2 – 2x – 3.
5x(x + 1)(x – 3) is completely factored.

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

7. Check It Out! Example 1
Tell whether each expression is completely factored. If not, factor it.
A. 5x2(x – 1)
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.

8. To factor a polynomial completely, you may need to use more than one factoring method. Use the steps on the next slide to factor a polynomial completely.

9. Factoring Polynomials
Step 1 Check for a greatest common factor.
Step 2 Check for a pattern that fits the difference of two
squares or a perfect-square trinomial.
Step 3 To factor x2 + bx + c, look for two integers whose
sum is b and whose product is c.
To factor ax2 + bx + c, check integer factors of a and c in the binomial factors. The sum of the products of the outer and inner terms should be b.
Step 4 Check for common factors.

10. Additional Example 2A: Factoring by GCF and Recognizing Patterns
Factor 10x2 + 48x + 32 completely. Check your answer.
10x2 + 48x + 32
2(5x2 + 24x + 16)
Factor out the GCF.
2(5x + 4)(x + 4)
Factor 5x2 + 24x + 16.
Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16)
= 10x2 + 40x + 8x + 32
= 10x2 + 48x + 32 

11. Additional Example 2B: Factoring by GCF and Recognizing Patterns
Factor 8x6y2 – 18x2y2 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 = 2x2, b = 3
Check 2x2y2(2x2 – 3)(2x2 + 3) = 2x2y2(4x4 – 9)
= 8x6y2 – 18x2y2 

12. Check It Out! Example 2a
Factor each polynomial completely. Check your answer.
4x3 + 16x2 + 16x
Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2.
4x3 + 16x2 + 16x
4x(x2 + 4x + 4)
4x(x + 2)2
a = x, b = 2
Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4)
= 4x(x2 + 4x + 4)
= 4x3 + 16x2 + 16x 

13. Check It Out! Example 2b
Factor each polynomial completely. Check your answer.
2x2y – 2y3
Factor out the GCF. 2y(x2 – y2) is a perfect-square trinomial of the form a2 – b2.
2x2y – 2y3
2y(x2 – y2)
2y(x + y)(x – y)
a = x, b = y
Check 2y(x + y)(x – y) = 2y(x2 – xy + xy – y2)
= 2x2y – 2xy2 + 2xy2 – 2y3
= 2x2y – 2y3 

14. For a polynomial of the form ax2 + bx + c, if there are no integers whose sum is b and whose product is ac, then the polynomial is said to be unfactorable.
Helpful Hint

15. Additional Example 3A: Factoring by Multiple Methods
Factor each polynomial completely.
9x2 + 3x – 2
The GCF is 1 and there is no pattern.
9x2 + 3x – 2
( 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
3(–2) + 1(3) = –3
–1 and 2
3(2) + 3(–1) = 3 
(3x – 1)(3x + 2)

16. Additional Example 3B: Factoring by Multiple Methods
Factor each 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 b = 2
12b(b + 2)(b + 2)
12b(b + 2)2

17. Additional Example 3C: Factoring by Multiple Methods
Factor each 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)

18. Additional Example 3D: Factoring by Multiple Methods.
Factor each polynomial completely.
(x4 – x2)
x2(x2 – 1)
Factor out the GCF.
x2(x + 1)(x – 1)
x2 – 1 is a difference of two squares.

19.   Check It Out! Example 3a Factor each polynomial completely.
3x2 + 7x + 4
a = 3 and c = 4;
Outer + Inner = 7
3x2 + 7x + 4
( x + )( x + ) 
Factors of 3 Factors of 4 Outer + Inner
3 and 1
1 and 4
3(4) + 1(1) = 13
2 and 2
3(2) + 1(2) = 8
4 and 1
3(1) + 1(4) = 7 
(3x + 4)(x + 1)

20.  Check It Out! Example 3b Factor each 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
– 1 and
The factors needed are –1 and 6.
2p3(p + 6)(p – 1)

21.   Check It Out! Example 3c Factor each polynomial completely.
9q6 + 30q5 + 24q4
Factor out the GCF. There is no pattern.
3q4(3q2 + 10q + 8)
( q + )( q + )
a = 3 and c = 8;
Outer + Inner = 10 
Factors of 3 Factors of 8 Outer + Inner
3 and 1
1 and 8
3(8) + 1(1) = 25
2 and 4
3(4) + 1(2) = 14
4 and 2
3(2) + 1(4) = 10 
3q4(3q + 4)(q + 2)

22. Check It Out! Example 3d
Factor each polynomial completely.
2x4 + 18
2(x4 + 9)
Factor out the GFC.
x4 + 9 is the sum of squares and that is not factorable.
2(x4 + 9) is completely factored.

24. Lesson Quiz
Tell whether the polynomial is completely factored. If not, factor it.
1. (x + 3)(5x + 10) x2(x2 + 9)
no; 5(x+ 3)(x + 2)
completely factored
Factor each polynomial completely. Check your answer.
3. x3 + 4x2 + 3x x2 + 16x – 48
4(x + 6)(x – 2)
(x + 4)(x2 + 3)
5. 18x2 – 3x – 3
6. 18x2 – 50y2
3(3x + 1)(2x – 1)
2(3x + 5y)(3x – 5y)
7. 5x – 20x3 + 7 – 28x2
(1 + 2x)(1 – 2x)(5x + 7)