Post on 03-Jan-2016
Topic 2Factoring Trinomials of the Form
Unit 7 Topic 2
InformationTo multiply two binomials you need to apply the distributive property twice.
For example, to multiply you need to multiply a by . Then, you need to multiply b by . In total, you need to perform four multiplications.
You can use the acronym FOIL or use a multiplication box to keep track of the four products. Both approaches are shown.
a b c d
c d c d
Information
InformationMultiplication BoxDraw a two-by-two box. Along one side, write the terms of the first binomial. Along another side, write the terms of the second binomial. In the cells of the box write the product of each pair of terms.
ExploreInvestigating Multiplying Binomials and Factoring Trinomials
1. Multiply the following factors so that the expression is in the form of .
a)
b)
c)
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2ax bx c
2 3
2 5
1 3
x x
x x
x x
ExploreInvestigating Multiplying Binomials and Factoring Trinomials
1. Multiply the following factors so that the expression is in the form of .
a) b)
Try this on your own first!!!!
2ax bx c
2
2
2 3
3 3
2
5
2
3 6
6
2x
x x
x
x x x
x x
x x
2
2
2 5
5 5
5 2 10
2
1
2
7 0
x x x
x
xx x
x
x x
x
Collect the like terms
ExploreInvestigating Multiplying Binomials and Factoring Trinomials
c)
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2
2
1 3
3 3
1
4
1
3 3
3
1x
x x
x
x x x
x x
x x
Explore
2. Use inductive reasoning to create a conjecture about how to determine b, when rewriting in the form . (Hint: Look at the original factors and the mathematical operations applied)
3. Use inductive reasoning to create a conjecture about how to determine c, when rewriting in the form . (Hint: Look at the original factors and the mathematical operations applied)
2ax bx c
2ax bx c
Collect the like terms by determining the sum of the two numbers = b
Multiply the two numbers to obtain a product = c
Example 1Solving Using the Zero Product Rule
Find the roots to the following quadratic equations. a)
b)
c)
d)
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2
6 2 0
3 1 9 0
3 2 4 0
2 1 0
x x
x x
x x
x
Example 1: SolutionsSolving Using the Zero Product Rule
Find the roots to the following quadratic equations. a) b)
6 0 2
6 2
0
6 2
6,
0
2
x x
x x
x
x x
1 0 9
3 1 9
9
,
0
0
1
1 9
x x
x x
x
x
x
Example 1: Solutions
c) d)
3 2 0 4 0
2, 4
3 4 0
3
2
x x
x
x x
22
2 1 0
12
1 0
x
x
x
More InformationA polynomial with three terms is called a trinomial. For example, is a trinomial.
To factor a trinomial of the form , find two numbers, r and s, with a product of the constant term, c, and a sum of the coefficient of the x term, b. If rs = c and r + s = b, then Below is the factorization of .
2 5 6x x
2x bx c
2x bx c x r x s 2 5 6x x
Example 2Factoring Trinomials of the Form
Identify two integers with the given product and sum.a) product = 18; sum =11
b) product = 12; sum = -7
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2 0x bx c
Example 2:Solutionsa) product p= 18 b) product p= 12
sum s=11 sum s= -7
1 18 18 1 18 19
2 9 18 2 9 11
3
P S
6 18 3 6 9
4
5
intege
6 3 18
rs a
6 3
re 2, 9
9
1 12 12 1 12 13
2 6 12 2 6 8
3
integers are 3
4 12 3 4 7
4 3 1
, 4
2
Positive product and a negative sum indicates that both integers are negative
Products:
Example 3Factoring Trinomials of the Form
Factor and determine the roots.a)
b)
c)
d)
e)
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2
2
2
2
2
11 18 0
7 12 0
20 0
5 36 0
10 25 0
x x
x x
x x
x x
x x
2 0x bx c
Example 3a: Solution
Since the c-value is +, and the b-value is +, we know that both signs will be positive.
sum product
Then, set each factor equal to zero and solve for the variable.
2
( 2)( 9) 0
2 0 9 0
1 0
9
1 8
,
1
2
x
x
x
x
x
x
x1 18 18 1 18 19
2 9 18 2 9 11
3
P S
6 18 3 6 9
4
5
intege
6 3 18
rs a
6 3
re 2, 9
9
Products:
Example 3a: SolutionTo check substitute each of our solutions into the equation to make sure they work.
2
Left Hand Side (LHS) Right Hand Side (RHS
( 2) 11( 2) 18 0
4 22 18
0 0
2 11 18 0x x
2
Left Hand Side (LHS) Right Hand Side (RHS
( 9) 11( 9) 18 0
81 99 18
0 0
Example 3b: Solution
Since the c-value is +, and the b-value is -, we know that both signs will be negative.
sum product
Then, set each factor equal to zero and solve for the variable.
2
( 3)( 4
7 12
) 0
3 0 4 0
3, 4
0
x x
x x
x
x x
Products:
1 12 12 1 12 13
2 6 12 2 6 8
3 4
inte
12
gers are
3 4 7
4 3 12
4
P
,
S
3
Example 3c: SolutionSince the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive the larger integer will have the positive sign.
sum product
Then, set each factor equal to zero and solve for the variable.
2
( 4)( 5) 0
4 0 5 0
2
, 5
0
4
0
x x
x
x
x
x
x
Products:
1 20 20 1 20 19
P
2 10 20 2 10
3
S
8
inte
4 5
ge
20
rs
4 5
ar 5
1
e 4,
Example 3d: SolutionSince the c-value is -, and the b-value is -, we know that one sign will be positive and one sign will be negative. Because the sum is negative the larger integer will have the negative sign.
sum product
Then, set each factor equal to zero and solve for the variable.
2
( 4)( 9)
5 36
0
4 0 9 0
4,
0
9
x x
x
x
x
x
x
Products:
1 36 36 1 36 35
2 18 36 2 18 16
3 12
integers are
36 3 12 9
4 9 36 4 9
9
S
,
P
4
5
Example 3e: Solution
Since the c-value is +, and the b-value is +, we know that both signs will be positive.
sum product
Then, set each factor equal to zero and solve for the variable.
2
2
( 5)( 5) 0 or ( 5)
10 2
0
5
5 0
0
5
x x
x x x
x
x
1 25 25 1 25 6
P
2
2
S
34
integ
5 5 2
ers
5 5 5 1
are 5, 5
0
Products:
Example 4Solving Quadratic Equations by Factoring
Factor and then solve each of the following quadratic equations.a)
b)
c)
d)
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2
2
2
2
2 4 30 0
3 21 36 0
5 10 15 0
7 28 21 0
x x
x x
x x
x x
Helpful Hint
When factoring, check for common factors first.
Example 4a: Solution
sum product
Then, set each factor equal to zero and solve for the variable.
2
2
( 2 15) 0
( 5)( 3)
2 4 30
0
5 0 3 0
2 2
2
0
5
2
2
, 3
x x
x x
x
x
x
x
x
1 15 15 1 15 14
2
P S
inte
3 5
ge
15
rs
3 5
ar 5
2
e -3,
Products:
Since the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive the larger integer will have the positive sign.
When trinomials look like they have an a-value that is not 1, look for a common factor that can be factored out.
Example 4a: SolutionTo check substitute each of our solutions into the equation to make sure they work.
2
Left Hand Side (LHS) Right Hand Side (RHS
2( 5) 4( 5) 30 0
50 20 30
0 0
22 4 30 0x x
2
Left Hand Side (LHS) Right Hand Side (RHS
2(3) 4(3) 30 0
18 12 30
0 0
Example 4b: Solution
sum product
Then, set each factor equal to zero and solve for the variable.
2
2
3 3 3
3
3
( 7 12) 0
( 3)( 4) 0
3 0 4
3 21 36
,
0
0
3 4
x x
x x
x x
x x
x
1 12 12 1 12 13
2 6 12
P
2 6 8
integers are 3
3 4 12 3
, 4
S
4 7
Products:
When trinomials look like they have an a-value that is not 1, look for a common factor that can be factored out.
Since the c-value is +, and the b-value is -, we know that both signs will be negative.
Example 4c: Solution
sum product
Then, set each factor equal to zero and solve for the variable.
2
2( 2 3) 0
( 1)( 3) 0
1 0 3 0
1,
5 5 5
5
5 10 1
3
0
5
5x x
x x
x x
x x
x
integers ar
1
e
3 3
P
1 3 2
1,
S
3
Products:
When trinomials look like they have an a-value that is not 1, look for a common factor that can be factored out.
Since the c-value is -, and the b-value is +, we know that one sign will be positive and one sign will be negative. Because the sum is positive, the larger integer will have the positive sign.
Example 4d: Solution
sum product
Then, set each factor equal to zero and solve for the variable.
2
2
7 7 7
7
7
7 28 21
( 4 3) 0
( 1)( 3) 0
1 0 3 0
0
1, 3
x x
x x
x
x
x
x
x
in
P S
tegers are 1
1 3
3
1 3 4
,
3
Products:
When trinomials look like they have an a-value that is not 1, look for a common factor that can be factored out.
Since the c-value is +, and the b-value is -, we know that both signs will be negative.
Example 5Determining the Dimensions of a Ping Pong Table
The area of a rectangular Ping Pong table is 45 ft2. The length is 4 feet longer than the width. a) If w represents the width of the table, then write an
expression for the length.
b) Substitute into the area formula A = lw and solve for the width. Start by rewriting the equation with the left side equal to zero.
Try this on your own first!!!!Try this on your own first!!!!
Example 5Determining the Dimensions of a Ping Pong Table
The area of a rectangular Ping Pong table is 45 ft2. The length is 4 feet longer than the width. a) If w represents the width of the table, then write an
expression for the length.
width
4
w
l w
Example 5The area of a rectangular Ping Pong table is 45 ft2. The length is 4 feet longer than the width. b) Substitute into the area formula A = lw and solve
for the width. Start by rewriting the equation with the left side equal to zero.
2
2
45
45 ( )
4
0 4 45
4
45 45
w w
w
l
w
A w
w
w
4l w
1 45 12 1 45 4
2
P S
4
3 15 12 3 15 12
4
integ
5
ers are 5
9 12 9 4
,
5
9
20 4 45
0 ( 5)( 9)
5, 9
w w
w w
w
Example 5Determining the Dimensions of a Ping Pong Table
c) What are the dimensions of a Ping Pong Table?
width
4
w
l w
20 4 45
0 ( 5)( 9)
5, 9
w w
w w
w
w is the width of the ping pong table so only positive values make sense in this context
5
4
5 4
9
w
l w
l
l
The dimensions of the Ping Pong table are a width of 5 feet and a length of 9 feet.
Need to Know:• To factor a trinomial of the form , find two
numbers, r and s, with a product of the constant term, c, and a sum of the coefficient of the x term, b. If rs = c and r + s = b, then
• When trinomials look like they have an a-value that is not 1, determine whether or not a common factor can be factored out.
• You can solve some quadratic equations by factoring. First write the equation in the form , with one side of the equation equal to zero. Then factor the other side. Next, set each factor to zero, and solve for the unknown.
You’re ready! Try the homework from this section.
2x bx c
2 ( )( )x bx c x r x s
2 0ax bx c