Factor quadratic trinomials of the form ax2 bx c'

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Objective
Factor quadratic trinomials of the form ax2 bx
c.
In the previous lesson you factored trinomials of
the form x2 bx c. Now you will factor
trinomials of the form ax bx c, where a ? 0.
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When you multiply (3x 2)(2x 5), the
coefficient of the x2-term is the product of the
coefficients of the x-terms. Also, the constant
term in the trinomial is the product of the
constants in the binomials.
(3x 2)(2x 5) 6x2 19x 10
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Write two numbers that are factors of a next to
the xs and two numbers that are factors of c in
the other blanks. Multiply the binomials to see
if you are correct.
(3x 2)(2x 5) 6x2 19x 10
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Example 1 Factoring ax2 bx c by Guess and
Check
Factor 6x2 11x 4 by guess and check.
Write two sets of parentheses.
The first term is 6x2, so at least one variable
term has a coefficient other than 1.
The coefficient of the x2 term is 6. The constant
term in the trinomial is 4.
Try factors of 6 for the coefficients and factors
of 4 for the constant terms.
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Example 1 Continued
Factor 6x2 11x 4 by guess and check.
Write two sets of parentheses.
The first term is 6x2, so at least one variable
terms has a coefficient other than 1.
The factors of 6x2 11x 4 are (3x 4) and (2x
1).
6x2 11x 4 (3x 4)(2x 1)
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Check It Out! Example 1a
Factor each trinomial by guess and check.
6x2 11x 3
Write two sets of parentheses.
The first term is 6x2, so at least one variable
term has a coefficient other than 1.
The coefficient of the x2 term is 6. The constant
term in the trinomial is 3.
Try factors of 6 for the coefficients and factors
of 3 for the constant terms.
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Check It Out! Example 1a Continued
Factor each trinomial by guess and check.
6x2 11x 3
Write two sets of parentheses.
The first term is 6x2, so at least one variable
term has a coefficient other than 1.
The factors of 6x2 11x 3 are (3x 1)(2x
3).
6x2 11x 3 (3x 1)(2x 3)
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Check It Out! Example 1b
Factor each trinomial by guess and check.
3x2 2x 8
Write two sets of parentheses.
The first term is 3x2, so at least one variable
term has a coefficient other than 1.
The coefficient of the x2 term is 3. The constant
term in the trinomial is 8.
Try factors of 3 for the coefficients and factors
of 8 for the constant terms.
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Check It Out! Example 1b
Factor each trinomial by guess and check.
3x2 2x 8
Write two sets of parentheses.
The first term is 3x2, so at least one variable
term has a coefficient other than 1.
The factors of 3x2 2x 8 are (x 2)(3x 4).
3x2 2x 8 (x 2)(3x 4)
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So, to factor a2 bx c, check the factors of a
and the factors of c in the binomials. The sum of
the products of the outer and inner terms should
be b.
( X )( x ) ax2 bx c
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Since you need to check all the factors of a and
the factors of c, it may be helpful to make a
table. Then check the products of the outer and
inner terms to see if the sum is b. You can
multiply the binomials to check your answer.
( X )( x ) ax2 bx c
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Example 2A Factoring ax2 bx c When c is
Positive
Factor each trinomial. Check your answer.
2x2 17x 21
a 2 and c 21, Outer Inner 17.
Use the Foil method.
(x 7)(2x 3)
Check (x 7)(2x 3) 2x2 3x 14x 21
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When c is negative, one factor of c will be
positive and the other factor will be negative.
Only some of the factors are shown in the
examples, but you may need to check all of the
possibilities.
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Example 2B Factoring ax2 bx c When c is
Positive
Factor each trinomial. Check your answer.
3x2 16x 16
a 3 and c 16, Outer Inner 16.
(x 4)(3x 4)
Use the Foil method.
Check (x 4)(3x 4) 3x2 4x 12x 16
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Check It Out! Example 2a
Factor each trinomial. Check your answer.
6x2 17x 5
a 6 and c 5, Outer Inner 17.
Use the Foil method.
(3x 1)(2x 5)
Check (3x 1)(2x 5) 6x2 15x 2x 5
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Check It Out! Example 2b
Factor each trinomial. Check your answer.
9x2 15x 4
a 9 and c 4, Outer Inner 15.
(3x 4)(3x 1)
Use the Foil method.
Check (3x 4)(3x 1) 9x2 3x 12x 4
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Check It Out! Example 2c
Factor each trinomial. Check your answer.
3x2 13x 12
a 3 and c 12, Outer Inner 13.
(x 3)(3x 4)
Use the Foil method.
Check (x 3)(3x 4) 3x2 4x 9x 12
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Example 3A Factoring ax2 bx c When c is
Negative
Factor each trinomial. Check your answer.
3n2 11n 4
a 3 and c 4, Outer Inner 11 .
(n 4)(3n 1)
Use the Foil method.
Check (n 4)(3n 1) 3n2 n 12n 4
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Example 3B Factoring ax2 bx c When c is
Negative
Factor each trinomial. Check your answer.
2x2 9x 18
a 2 and c 18, Outer Inner 9.
Use the Foil method.
Check (x 6)(2x 3) 2x2 3x 12x 18
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Example 3C Factoring ax2 bx c When c is
Negative
Factor each trinomial. Check your answer.
4x2 15x 4
a 4 and c 4, Outer Inner 15.
Use the Foil method.
Check (x 4)(4x 1) 4x2 x 16x 4
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Check It Out! Example 3a
Factor each trinomial. Check your answer.
6x2 7x 3
a 6 and c 3, Outer Inner 7.
Use the Foil method.
Check (3x 1)(2x 3) 6x2 9x 2x 3
6x 7x 3
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Check It Out! Example 3b
Factor each trinomial. Check your answer.
4n2 n 3
a 4 and c 3, Outer Inner 1.
Use the Foil method.
Check (4n 3)(n 1) 4n2 4n 3n 3
4n2 n 3
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When the leading coefficient is negative, factor
out 1 from each term before using other
factoring methods.
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Example 4A Factoring ax2 bx c When a is
Negative
Factor 2x2 5x 3.
1(2x2 5x 3)
Factor out 1.
a 2 and c 3 Outer Inner 5
1(x 1)(2x 3)
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Check It Out! Example 4a
Factor each trinomial. Check your answer.
6x2 17x 12
Factor out 1.
1(6x2 17x 12)
a 6 and c 12 Outer Inner 17
1(2x 3)(3x 4)
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Check It Out! Example 4b
Factor each trinomial. Check your answer.
3x2 17x 10
Factor out 1.
1(3x2 17x 10)
a 3 and c 10 Outer Inner 17)
1(3x 2)(x 5)