Lesson 8-6 Warm-Up

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Lesson 8-6 Warm-Up
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Factoring Trinomials of the Type ax2 bx c
(8-6)

• How do you factor a trinomial in which the second
  degree variable has a coefficient?

• To factor a trinomial of the form ax2 bx c,
  you must find two factor pairs of a and c whose
  sum is b. There are two methods to factoring a
  trinomial in the form of ax2 bx c.
• Method 1 Create a Three Column Table Title one
  column Factors of First Term, the second column
  Sum of the Factors, and the last column
  Factors of Third Term. Then, fill in the table
  and choose the products whose sum is the middle
  number, b.
• Example Factor 6n2 23n 7

Factors of 3rd Term
1 and 7 7
1 and 7 7
1 and 7 7
Sum of the Factors
11 67 ? 23
17 61 ? 23
21 37 23 ?
Factors of 1st Term
1 and 6 6
1 and 6 6
2 and 3 6
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Factoring Trinomials of the Type ax2 bx c
(8-6)

• So, use the factors 2 and 3 for the first terms
  of the binomials and use 1 and 7 for the second
  terms of the binomials. Now, we need to figure
  out what order to put those combinations in.
• (2n 1)(3n 7)
• Check (using FOIL) (2n 1)(3n 7) 6n2 14n
  3n 7 6n2 17n 7
• ? 6n2 23n 7
• Since this combination doesnt work, switch the
  first of last terms. Below, the last terms are
  switched.
• (2n 7)(3n 1)
• Check (using FOIL) (2n 7)(3n 1) 6n2 21n
  2n 7 6n2 23n 7 ?

4
Factoring Trinomials of the Type ax2 bx c
(8-6)

• Method 2 Use the FOIL Method in Reverse Look
  for binomials that have the following
  characteristics for ax2 bx c.
• Example Factor 6n2 23n 7

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Factoring Trinomials of the Type ax2 bx c
(8-6)

• Method 3 Use an Area Model in Reverse Arrange
  the Algebra Tiles that model the trinomial into a
  rectangle. The sides of the rectangle (length and
  width) are the factors of the trinomial. Tip
  Think about how to end with the number of 1
  tiles.
• Example Factor 6n2 23n 7

2n 7
Algebra Tiles Key
n2
n2
n
n
n
n
n
n
n
1
n
1
1
n
1
n2
n2
n
n
n
n
n
n
n
3n 1
3n 1
n
n
n2
n
n2
n2
n
n
n
n
n
n
n
n
1
n
n
1
1
1
1
1
1
1
2n 7
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Factoring Trinomials of the Type ax2 bx c
(8-6)
Method 4 Use an Area Model in Reverse (X-Box
Method)

1. Find two numbers whose product is ac and sum is
   b. These numbers will be the coefficients of the
   x terms.

1. Then, create a box divided into two columns and
   two rows. The top-left box will be the a term,
   the bottom right box will be the c term, and the
   middle two boxes will be the b terms.

1. Finally, find common factors of each column and
   row. The dimensions (length and width) of the box
   are factors (binomial times binomial) of the
   trinomials.

• Example Factor 6n2 23n 7

2n
7
3n
6n2
21n
42n2
2n
21n
7
2n
1
23
Answer (2n 7)(3n 1)
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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Additional Examples
Factor 20x2 17x 3.
Method 1 Table
Factors of 1st Term Sum of the Factors Factors of 3rd Term
20 and 1 20 201 13 ? 17 1 and 3 3
20 and 1 20 203 11 ? 17 1 and 3 3
10 and 2 20 101 23 ? 17 1 and 3 3
10 and 2 20 103 21 ? 17 1 and 3 3
5 and 4 20 51 43 17 ? 1 and 3 3
5 and 4 20 53 41 ? 17 1 and 3 3
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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Additional Examples
So, use the factors 5 and 4 for the first terms
of the binomials and use 1 and 3 for the second
terms of the binomials. Now, we need to figure
out what order to put those combinations. (5x
1)(4x 3) Check (using FOIL) (5x 1)(4x
3) 20x2 15x 4x 3 20x2 19x 3 ?
20x2 17x 3 Since this combination doesnt
work, switch the first of last terms. Below, the
last terms are switched. (5x 3)(4x
1) Check (FOIL) (5x 1)(4x 3) 20x2 5x
12x 3 20x2 17x 3 ?
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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Additional Examples
Factor 20x2 17x 3.
Method 2 FOIL
20x 17x 3
F O I L
2 10 2 3 1 10 16 1 3 2 10 2 1
3 10 32 3 1
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Factoring Trinomials of the Type ax2 bx c
(8-6)
Factor 20x2 17x 3.
Method 3 X-Box Method

• Factor 20x2 17x 3

5x
3
60x2
4x
20x2
12x
5x
12x
5x
17
3
1
Answer (5x 3)(4x 1)
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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Additional Examples
Factor 3n2 7n 6.
3n2 7n 6
F O I
L
(1)(3)     (1)(6) (3)(1) 3 (1)(6)
(1)(3)  (1)(1) (3)(-6) 17 (-6)(1)
(1)(3)  (1)(-3) (3)(2) 3 (2)(3)
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Factoring Trinomials of the Type ax2 bx c
(8-6)

• Note Some polynomials have terms with a common
  factor. If this is the case, factor out that
  monomial factor using the Distributive Property
  in reverse before factoring the trinomial.
• Example Factor 20x2 80x 35
• Step 1 20x2 80x 35 5(4x2 16x 7) 5 is a
  common factor of all 3 terms, so factor it
  out.
• Step 2 Factor 4x2 16x 7.
• Step 3 Find the correct combination of the
  factor pairs 2, 2 and 1,7 to equal 4x2 16x 7.
  
• (2x 1)(2x 7)
• Check (FOIL) (2x 1)(2x 7) 4x2 14x 2x
  7 4x2 16x 7 ?

Factors of 1st Term Sum of the Factors Factors of 3rd Term
1 and 4 4 11 47 ? 16 1 and 7 7
1 and 4 4 17 41 ? 16 1 and 7 7
2 and 2 4 21 27 16 ? 1 and 7 7
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Factoring Trinomials of the Type ax2 bx c
(8-6)

• Dont forget to multiply the binomials by the
  factors you pulled out (in other words, put the
  common factor in front of the answer).
• 5(2x 1)(2x 7)
• So, the 20x2 80x 35 completely factored is
  5(2x 1)(2x 7).

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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Additional Examples
Factor 18x2 33x 30 completely.
18x2 33x 30 3(6x2 11x 10) Factor out
the common factor.
Factor 6x2 11x 10.
F O I
L
6x2 11x 10
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Factoring Trinomials of the Type ax2 bx c
LESSON 8-6
Lesson Quiz
Factor each expression. 1. 3x2 14x
11 2. 6t2 13t 63 3. 9y2 48y 36
(x 1)(3x 11)
(2t 9)(3t 7)
3(3y 2)(y 6)