topic 2 unit 7 topic 2. information to multiply two binomials you need to apply the distributive...

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)

Try this on your own first!!!!Try this on your own first!!!!

2ax bx c

2 3

2 5

1 3

x x

x x

x x


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


c)

Try this on your own first!!!!

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)


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


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)


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


( 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


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


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


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


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)


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


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


2( 5) 4( 5) 30 0

50 20 30

0 0

22 4 30 0x x

2


2(3) 4(3) 30 0

18 12 30

0 0

Example 4b: Solution

sum product


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:



Example 4c: Solution

sum product


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:


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


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:



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.



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


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

topic 2 unit 7 topic 2. information to multiply two binomials you need to apply the distributive...

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