8-6
Factoring ax2 + bx + c
Vocabulary
Review
Complete each sentence with the correct form of the word factor.
factor
factored
factoring
1. The monomials 12x, 3x 2, and 6x 3 have a common 9 of 3x.
2. When 9 a trinomial, begin by looking for a factor common to all three terms.
3. A student 9 the expression 12x 3 2 6x 2 into 6x 2(2x 2 1).
Vocabulary Builder
rewrite (verb) ree ryt
Main Idea: Sometimes you can rewrite trinomials in a different, but equivalent,
way to make them easier to factor.
Definition: Rewrite means to write in another form.
Use Your Vocabulary
4. You can rewrite the expression 23x 1 14 1 3x 2 in order to factor it. Cross out the
rewritten expression that is NOT equivalent to the original expression.
3x 2 1 23x 1 14
3x 2 1 2x 1 21x 1 14
3x 2 1 20x 1 2x 1 14
5. Circle the words that are similar in meaning to rewrite.
revise
restate
compose
6. Draw a line from each trinomial in Column A to an equivalent and rewritten
expression in Column B.
Column A
Column B
a2 1 2ab 1 b2 a2 1 6ab 1 b2
a2 1 4ab 1 b2 a2 1 ab 1 ab 1 b2
b2 1 a2 1 6ab
a2 1 b2 1 4ab
Chapter 8
246 change Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Other Word Form: rewritten
Problem 1 Factoring When ac Is Positive
Got It? What is the factored form of 6x2 1 13x 1 5?
7. Complete the diagram below.
6x 2 + 13x + 5
= 30
•
8. Circle the factors of 30 that sum to 13.
1 and 30 2 and 15
3 and 10
HSM11_A1MC_0806_T13404
21 and 230
22 and 215
23 and 210
5 and 6
25 and 26
9. Reasoning Why is the pair 23 and 10 not one of the pairs listed in Exercise 8?
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10. Use your answer to Exercise 8 to complete the diagram below. Then rewrite
the expression.
+
6x 2
+
13x
5
+
=
6x2 + 3x
+
+
)+5(
= 3x (
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
3
+
+
)
3
The expressions inside the parentheses must be equal.
Use the Distributive Property to factor out
the GCF, the part inside the parentheses.
= (3x + 5)(
2
+
)
11. Circle the
factored form of 6x2 1 13x 1 5.
HSM11_A1MC_0806_T13405
6x 2 1 13x 1 5
6x 2 1 3x 1 10x 1 5
(6x 2 1 3x)(10x 1 5)
(3x 1 5)(2x 1 1)
12. Reasoning Suppose you rewrite the expression as 6x 2 1 10x 1 3x 1 5. How
would the GCF change? Explain.
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________________________________________________________________________
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247
Lesson 8-6
Problem 2 Factoring When ac Is Negative
Got It? What is the factored form of 10x2 1 31x 2 14?
13. Underline the correct word to complete each sentence.
The product ac is negative / positive .
The sum b of the factors of ac is negative / positive .
The factor of ac with the greater absolute value must be negative / positive .
The factor of ac with the lesser absolute value must be negative / positive .
14. Write the factor pairs of ac = 2140. Then circle the factor pair that sums to 31.
15. Use the justifications at the right to factor the expression.
10x 2 1 31x 2 14 5 x 2 2 5 ( 5 ( x 1 x 2 1 )( x 2 14
) 1 ( x 2 )
Rewrite bx.
x 2 Factor the GCF from
)each pair of terms.
Use the Distributive Property.
Problem 3 Applying Trinomial Factoring
Got It? The area of a rectangle is 8x 2 1 22x 1 15. What are possible dimensions
of the rectangle? Use factoring.
17. The factor pairs of ac are shown below. Circle the factor pair that sums to b.
1 and 120
2 and 60
5 and 24
3 and 40
6 and 20
4 and 30
8 and 15
10 and 12
18. Use the justifications below to factor the trinomial.
8x 2 1 x 1 x 1 15
( ) 1 ( )( ( Rewrite bx.
)
)
19. Possible dimensions of the rectangle are Chapter 8
Factor the GCF from each pair of terms.
Use the Distributive Property.
and 248 .
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16. Check your work.
Problem 4 Factoring Out a Monomial First
Got It? What is the factored form of 8x2 2 36x 2 20?
20. Circle the first step in factoring the expression. Underline the second step.
Find factors of the product ac.
Factor out the GCF.
Find the product ac. 21. The polynomial is factored below. Write a justification for each step.
8x 2 2 36x 2 20 5 4(2x 2 2 9x 2 5)
5 4(2x 2 2 10x 1 1x 2 5)
5 4f2x(x 2 5) 1 1(x 2 5)g 5 4(2x 1 1)(x 2 5)
Lesson Check • Do you UNDERSTAND?
Reasoning Explain why you cannot factor the trinomial 2x 2 1 7x 1 10.
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22. Consider the polynomial 2x 2 1 9x 1 10.
Can you factor out a monomial from the polynomial?
Yes / No Can you rewrite 9 as the sum of two factors of 20?
Yes / No Can you factor 2x 2 1 9x 1 10?
Yes / No 23. Now consider the polynomial
2x 2
1 7x 1 10.
Can you factor out a monomial from the polynomial?
Yes / No Can you rewrite 7 as the sum of two factors of 20?
Yes / No 24. Complete. I cannot factor 2x 2 1 7x 1 10 because 9.
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Math Success
Check off the vocabulary words that you understand.
factoring
trinomial
standard form
Rate how well you can factor trinomials of the form ax 2 1 bx 1 c.
Need to
review
0
2
4
6
8
Now I
get it!
10
249
Lesson 8-6