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7.5 Factoring
Trinomials
CORD Math
Mrs. Spitz
Fall 2006
1
Objectives
• Factor quadratic trinomials.
2 of 14
Assignment
• Pg. 274 #5-43 all
3 of 14
Introduction
• In Lesson 7.1, you learned that when
two numbers are multiplied, each
number is a factor of the product.
Similarly, if two binomials are multiplied,
each binomial is a factor of the product.
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Consider the binomials 5x + 2 and 3x + 7. You
can use the FOIL method to find their products.
(5 x  2)(3x  7)  (5 x)(3x)  (5 x)(7)  2(3x)  2(7)
 15 x  35 x  6 x  14
2
 15 x 2  (35  6) x  14
 15 x  41x  14
2
The binomials 5x + 2 and 3x + 7 are factors of
15x2 + 41x + 14.
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FOIL
• When using the FOIL method, look at
the product of the coefficients of the first
and last terms, 15 and 14. Notice that it
is the same as the product of the two
terms 35 and 6, whose sum is the
coefficient of the middle term. You can
use this pattern to factor quadratic
trinomials, such as 2y2 + 7y +6.
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Ex. 1: Factor
2
5x -
17x + 14
• The product of 5 and 14 is 70. Since the
product is positive and the sum is negative,
both factors must be negative.
• Possibilities:
Factors of 70
-1, -70
-2, -35
-5, -14
-7, -10
Sum of factors
-1+ -70 = -71 NOT
-2 + -35 = -37 NOT
-5 + -14 = -19 NOT
-7 + -10 = -17 YES
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Now what? Stop when you find the factors.
5 x 2  [10  (7)] x  14
5 x 2  10 x  7 x  14
(5 x 2  10 x)  (7 x  14)
5 x( x  2)  (7)( x  2)
Factor GCF from each
( x  2)(5 x  7)
Factor by grouping
Check using FOIL
( x  2)(5 x  7)
x 2  7 x  10 x  14
x 2  17 x  14
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Ex. 2: What does a factored trinomial look like?
What are the
factors of 6
- 1
+ 6
that subtract to
give you 5?
Look at the trinomial. Are the signs
(x
)(x
)
positive, negative or both positive and
negative. For example:
x2 + 5x - 6
6-1=5
One sign is positive, one negative meaning you are looking for
factors of 6 that subtract to give you 5—the number in the middle.
With no coefficient (the number in front of the variable x2 (the
letter), it’s pretty easy to figure out.
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Ex. 3: Factor
2
2n
-11n + 7
• You must find two numbers whose product is
2 ·7 or 14 and whose sum is -11.
Factors of 14
-1, -14
-2, -7
Sum of factors
-1+ -14 = -15
-2 + -7 = -9
There are no factors of 14 whose sum is -11.
Therefore this expression cannot be factored
using integers. It is prime.
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Ex. 4: Factor
2
7a
+ 22a + 3
• You must find two numbers whose product is
7 · 3 or 21 and whose sum is
-11.
Factors of 21
1, 21
3, 7
Sum of factors
1 + 21 = 22 YES
3 + 7 = 10 NO
Factors of 21 that add to give you 22 are 1 and
22. Factored format is
(7a+1)(a+3)
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a  22a  3
(7 a  1)( a  3)
FOIL
2
7 a  21a  1a  3
2
7 a  22a  3
Checks
2
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Ex. 5: Factor
2
2q
– 9q - 18
• You must find two numbers whose product is
2 · 18 or 36 and whose sum is -9.
Factors of 36
-36, 1
-18, 2
-12, 3
Sum of factors
-36+1=-35 NOT
-18 + 2 = -16 NOT
-12 + 3 = -9 YES
Factors of -36 that add to give you -9 are -12 and
3. Factored format is
(2q + 3)(q – 6)
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