Warm Up

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Preview
Warm Up
California Standards
Lesson Presentation
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Warm Up Find each product. 1. (x 2)(2x 7) 2.
(3y 4)(2y 9) 3. (3n 5)(n 7) Find each
trinomial. 4. x2 4x 32 5. z2 15z 36 6. h2
17h 72
2x2 3x 14
6y2 35y 36
3n2 26n 35
(x 4)(x 8)
(z 3)(z 12)
(h 8)(h 9)
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California Standards

11.0 Students apply basic factoring techniques to
second- and simple third- degree polynomials.
These techniques include finding a common factor
for all terms in a polynomial, recognizing the
difference of two squares, and recognizing
perfect squares of binomials.
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In the previous lesson you factored trinomials of
the form x2 bx c. Now you will factor
trinomials of the form ax2 bx c, where a ? 0
or 1.
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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 integers that are factors of a next to
the xs and two integers that are factors of c in
the other blanks. Then multiply to see if the
product is the original trinomial. If there are
no two such integers, we say the trinomial is not
factorable.
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Additional Example 1 Factoring ax2 bx c
Factor 6x2 11x 4. Check your answer.
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 integer factors of 6 for the coefficients and
integer factors of 4 for the constant terms.
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Additional Example 1 Continued
Factor 6x2 11x 4. Check your answer.
The factors of 6x2 11x 4 are (3x 4) and (2x
1).
6x2 11x 4 (3x 4)(2x 1)
(3x 4)(2x 1)
Check
Use the FOIL method.
6x2 3x 8x 4
?
6x2 11x 4
The product of the original trinomial.
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Check It Out! Example 1a
Factor each trinomial. Check your answer.
6x2 11x 3
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 integer factors of 6 for the coefficients and
integer factors of 3 for the constant terms.
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Check It Out! Example 1a Continued
Factor each trinomial. Check your answer.
The factors of 6x2 11x 3 are (3x 1)(2x
3).
6x2 11x 3 (3x 1)(2x 3)
Use the FOIL method.
6x2 9x 2x 3
?
6x2 11x 3
The product of the original trinomial.
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Check It Out! Example 1b
Factor each trinomial. Check your answer.
3x2 2x 8
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 integer factors of 3 for the coefficients and
integer factors of 8 for the constant terms.
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Check It Out! Example 1b
Factor each trinomial. Check your answer.
3x2 2x 8
The factors of 3x2 2x 8 are (x 2)(3x 4).
3x2 2x 8 (x 2)(3x 4)
Use the FOIL method.
3x2 4x 6x 8
?
The product of the original trinomial.
3x2 2x 8
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So, to factor ax2 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.
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Since you need to check all the factors of a and
all 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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Additional 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 FOILmethod.
(x 7)(2x 3)
Check (x 7)(2x 3) 2x2 3x 14x 21
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Additional 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
?
3x2 13x 12
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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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Additional 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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Additional 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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Additional 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 (2x 3)(3x 1) 6x2 2x 9x 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 (n 1)(4n 3) 4n2 3n 4n 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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Caution!
When you factor out 1 in an early step, you must
carry it through the rest of the steps and into
the answer.
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Additional Example 4 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)
Check 1(2x 3)(3x 4) 6x2 8x 9x 12
?
6x2 17x 12
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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(x 5)(3x 2)
Check 1(x 5)(3x 2) 3x2 2x 15x 10
?
3x2 17x 10
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Lesson Quiz
Factor each trinomial. Check your answer. 1. 5x2
17x 6 2. 2x2 5x 12 3. 6x2 23x
7 4. 4x2 11x 20 5. 2x2 7x 3 6. 8x2
27x 9
(5x 2)(x 3)
(2x 3)(x 4)
(3x 1)(2x 7)
(x 4)(4x 5)
(2x 1)(x 3)
(8x 3)(x 3)