9.6
Factor ax 2 1 bx 1 c
Goal
p Factor trinomials of the form ax2 1 bx 1 c.
Factor when b is negative and c is positive
Example 1
Your Notes
Factor 2x 2 2 11x 1 5.
Solution
Because b is negative and c is positive, both factors of
c must be
. You must consider the
of the factors of 5, because the x-terms of the possible
factorizations are different.
Factors
of 2
Factors
of 5
Possible
factorization
Middle term when
multiplied
1, 2
21,
(x 2 1)(2x
)
2 2x 5
1, 2
25,
(x 2 5)(2x
)
2 10x 5
2x 2 2 11x 1 5 5 (x 2
Example 2
)(2x
)
Factor when b is positive and c is negative
Factor 5n2 1 2n 2 3.
Solution
Because b is positive and c is negative, the factors of
c have
.
Factors
of 5
Factors
of 23
Possible
factorization
Middle term when
multiplied
1, 5
1,
(n 1 1)(5n
)
1, 5
21,
(n 2 1)(5n
)
1, 5
3,
(n 1 3)(5n
)
1, 5
23,
(n 2 3)(5n
)
5n2 1 2n 2 3 5 (n
Copyright © Holt McDougal. All rights reserved.
)(5n
)
Lesson 9.6 • Algebra 1 Notetaking Guide
237
9.6
Factor ax 2 1 bx 1 c
Goal
p Factor trinomials of the form ax2 1 bx 1 c.
Factor when b is negative and c is positive
Example 1
Your Notes
Factor 2x 2 2 11x 1 5.
Solution
Because b is negative and c is positive, both factors of
c must be negative . You must consider the order
of the factors of 5, because the x-terms of the possible
factorizations are different.
Factors
of 2
Factors
of 5
Possible
factorization
Middle term when
multiplied
1, 2
21,
25
(x 2 1)(2x
25 )
25x 2 2x 5 27x
1, 2
25,
21
(x 2 5)(2x
21 )
2x 2 10x 5 211x
2x 2 2 11x 1 5 5 (x 2 5 )(2x 2 1 )
Example 2
Factor when b is positive and c is negative
Factor 5n2 1 2n 2 3.
Solution
Because b is positive and c is negative, the factors of
c have different signs .
Factors
of 5
Factors
of 23
1, 5
1,
1, 5
21,
1, 5
3,
1, 5
23,
Possible
factorization
23 (n 1 1)(5n 2 3 )
Middle term when
multiplied
23n 1 5n 5 2n
3
(n 2 1)(5n
13 )
3n 2 5n 5 22n
21
(n 1 3)(5n
21 )
2n 1 15n 5 14n
1
(n 2 3)(5n
11 )
n 2 15n 5 214n
5n2 1 2n 2 3 5 (n 1 1 )(5n 2 3 )
Copyright © Holt McDougal. All rights reserved.
Lesson 9.6 • Algebra 1 Notetaking Guide
237
Your Notes
Checkpoint Factor the trinomial.
1. 3x 2 2 5x 1 2
Example 3
2. 2m2 1 m 2 21
Factor when a is negative
Factor 24x 2 1 4x 1 3.
Solution
Step 1 Factor
from each term of the trinomial.
24x 2 1 4x 1 3 5
(
. Because b
, the factors of c must
Step 2 Factor the trinomial
and c are both
have
Factors
of 4
Remember to
include the
that you factored
out in Step 1.
Factors
of 23
)
.
Possible
factorization
Middle term when
multiplied
1, 4
1,
(x 1 1)(4x
)
1, 4
3,
(x 1 3)(4x
)
1, 4
21,
(x 2 1)(4x
)
1, 4
23,
(x 2 3)(4x
)
2, 2
1,
(2x 1 1)(2x
)
2, 2
21,
(2x 2 1)(2x
)
24x 2 1 4x 1 3 5
Checkpoint Complete the following exercise.
3. Factor 22y 2 2 11y 2 5.
238
Lesson 9.6 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Factor the trinomial.
1. 3x 2 2 5x 1 2
2. 2m2 1 m 2 21
(x 2 1)(3x 2 2)
Example 3
(m 2 3)(2m 1 7)
Factor when a is negative
Factor 24x 2 1 4x 1 3.
Solution
Step 1 Factor 21 from each term of the trinomial.
24x 2 1 4x 1 3 5 2 ( 4x 2 2 4x 2 3 )
Step 2 Factor the trinomial 4x 2 2 4x 2 3 . Because b
and c are both negative , the factors of c must
have different signs .
Factors
of 4
Remember to
include the 21
that you factored
out in Step 1.
Factors
of 23
Possible
factorization
Middle term when
multiplied
1, 4
1,
23
(x 1 1)(4x
23 )
23x 1 4x 5 x
1, 4
3,
21
(x 1 3)(4x
21 )
2x 1 12x 5 11x
1, 4
21,
3
(x 2 1)(4x
13 )
3x 2 4x 5 2x
1, 4
23,
1
(x 2 3)(4x
11 )
x 2 12x 5 211x
2, 2
1,
23 (2x 1 1)(2x 2 3 )
26x 1 2x 5 24x
2, 2
21,
3
13 )
(2x 2 1)(2x
6x 2 2x 5 4x
24x 2 1 4x 1 3 5 2(2x 1 1)(2x 2 3)
Checkpoint Complete the following exercise.
3. Factor 22y 2 2 11y 2 5.
2(2y 1 1)(y 1 5)
238
Lesson 9.6 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 4
Write and solve a polynomial equation
Tennis An athlete hits a tennis ball at an initial height of
8 feet and with an initial vertical velocity of 62 feet per
second.
a. Write an equation that gives the height (in feet) of the
ball as a function of the time (in seconds) since it left
the racket.
b. After how many seconds does the ball hit the ground?
Solution
a. Use the
to write an equation
for the height h (in feet) of the ball.
h 5 216t 2 1 vt 1 s
h 5 216t 2 1
t1
v5
and s 5
b. To find the number of seconds that pass before the
ball lands, find the value of t for which the height
. Substitute
for h and solve the
of the ball is
equation for t.
5 216t 2 1
5
(
5
(
t1
Substitute
)
)(
for h.
Factor out
)
.
Factor the trinomial.
or
Zero-product property
or
Solve for t.
A negative solution does not make sense in this situation.
.
The tennis ball hits the ground after
Checkpoint Complete the following exercise.
Homework
4. What If? In Example 4, suppose another athlete
hits the tennis ball with an initial vertical velocity
of 20 feet per second from a height of 6 feet. After
how many seconds does the ball hit the ground?
Copyright © Holt McDougal. All rights reserved.
Lesson 9.6 • Algebra 1 Notetaking Guide
239
Your Notes
Example 4
Write and solve a polynomial equation
Tennis An athlete hits a tennis ball at an initial height of
8 feet and with an initial vertical velocity of 62 feet per
second.
a. Write an equation that gives the height (in feet) of the
ball as a function of the time (in seconds) since it left
the racket.
b. After how many seconds does the ball hit the ground?
Solution
a. Use the vertical motion model to write an equation
for the height h (in feet) of the ball.
h 5 216t 2 1 vt 1 s
Vertical motion
model
h 5 216t 2 1 62 t 1 8
v 5 62 and s 5 8
b. To find the number of seconds that pass before the
ball lands, find the value of t for which the height
of the ball is 0 . Substitute 0 for h and solve the
equation for t.
0 5 216t 2 1 62 t 1 8
Substitute 0 for h.
0 5 22 ( 8t 2 2 31t 2 4 )
Factor out 22 .
0 5 22 ( 8t 1 1 )( t 2 4 )
Factor the trinomial.
8t 1 1 5 0
Zero-product property
or t 2 4 5 0
1
8
t 5 2} or
t54
Solve for t.
A negative solution does not make sense in this situation.
The tennis ball hits the ground after 4 seconds .
Checkpoint Complete the following exercise.
Homework
4. What If? In Example 4, suppose another athlete
hits the tennis ball with an initial vertical velocity
of 20 feet per second from a height of 6 feet. After
how many seconds does the ball hit the ground?
1.5 seconds
Copyright © Holt McDougal. All rights reserved.
Lesson 9.6 • Algebra 1 Notetaking Guide
239