Download x - MDC Faculty Web Pages

Chapter 4
101.
Polynomials
5wx 3 + 5wy 3 ! 2zx 3 ! 2zy 3
= 5w(x 3 + y 3 ) ! 2z(x 3 + y 3 )
= (x 3 + y 3 )(5w ! 2z)
= (x + y)(x 2 ! xy + y 2 )(5w ! 2z)
Section 4.8
Solving Equations by Using the Zero Product Rule
Section 4.8 Practice Exercises
1.
a.
d.
If a, b, and c are real numbers such that a ≠ 0, then a quadratic equation is an equation that can
be written in the form ax 2 + bx + c = 0 .
The zero product rule states that if the product of two factors is zero, then one or both of its
factors is equal to zero.
Let a, b, and c represent real numbers such that a ≠ 0. Then a function in the form
f ( x ) = ax 2 + bx + c is called a quadratic function.
The graph of a quadratic function is called a parabola.
a.
x2 ! y2 = x ! y x + y
b.
x + y is prime.
b.
c.
2.
(
2
)(
)
2
)(
= ( x + y)( x
(
c.
x 3 ! y 3 = x ! y x 2 + xy + y 2
d.
x +y
3
3
(
2
! xy + y
2
)
)
)
3.
10x 2 + 3x = x (10x + 3)
4.
7x 2 ! 28 = 7 x 2 ! 4 = 7 ( x + 2 ) ( x ! 2 )
5.
2 p 2 ! 9 p ! 5 = 2 p 2 ! 10 p + p ! 5
= 2 p ( p ! 5) + ( p ! 5)
= ( p ! 5 ) ( 2 p + 1)
6.
3q 2 ! 4q ! 4 = 3q 2 ! 6q + 2q ! 4
= 3q ( q ! 2 ) + 2 ( q ! 2 )
= ( q ! 2 ) ( 3q + 2 )
7.
t 3 ! 1 = t 3 ! 13 = ( t ! 1) t 2 + t + 1
8.
z 2 ! 11z + 30 = z 2 ! 6z ! 5z + 30
= z ( z ! 6) ! 5 ( z ! 6)
= ( z ! 6)( z ! 5)
9.
The equation must be set equal to 0, and the
polynomial must be factored.
10.
If a ! b = 0 , then either a = 0!or!b = 0 .
11.
2x ( x ! 3) = 0 Correct form.
12.
(u + 1) (u ! 3) = 10 Incorrect form. The
equation is not set equal to 0.
13.
3p 2 ! 7 p + 4 = 0 Incorrect form. The
polynomial is not factored.
14.
t 2 ! t ! 12 = 0 Incorrect form. The
polynomial is not factored.
15.
a ( a + 3) = 5 Incorrect form. The equation
is not set equal to 0.
16.
1%
"2
%"
$# x ! 5 '& $# x + '& = 0 Correct form.
3
2
(
2
)
306
Section 4.8
17.
Solving Equations by Using the Zero Product Rule
( x + 3) ( x + 5 ) = 0
18.
( x + 7)( x ! 4 ) = 0
x + 3 = 0 or x + 5 = 0
x = !3 or
19.
x + 7 = 0 or x ! 4 = 0
x = !5!!!{!3,!!5}
x = !7 or
( 2w + 9 ) ( 5w ! 1) = 0
20.
( 3a + 1) ( 4a ! 5 ) = 0
2w + 9 = 0 or 5w ! 1 = 0
2w = !9 or 5w = 1
w=!
x = 4!!!{!7,!4}
3a + 1 = 0 or 4a ! 5 = 0
3a = !1 or
4a = 5
9
1 " 9 1%
or w = !!! #! ,! &
2
5 $ 2 5'
a=!
1
5 " 1 5%
or !!!!a = !!! #! ,! &
3
4 $ 3 4'
21.
x ( x + 4 ) (10x ! 3) = 0
x = 0 or x + 4 = 0 or 10x ! 3 = 0
x = 0 or
x = !4 or 10x = 3
3
x = 0 or
x = !4 or
x=
10
3%
"
!!!!!!!!!!!!! #0,!!4,! &
10 '
$
22.
t ( t ! 6 ) ( 3t ! 11) = 0
t = 0 or t ! 6 = 0 or 3t ! 11 = 0
t = 0 or
t = 6 or
3t = 11
11
t = 0 or
t = 6 or
t=
3
11 %
"
!!!!!!!!!!!!!!! #0,!6,! &
3'
$
23.
0 = 5 ( y ! 0.4 ) ( y + 2.1)
24.
0 = !4 ( z ! 7.5 ) ( z ! 9.3)
5 = 0 or y ! 0.4 = 0 or y + 2.1 = 0
no solution
y = 0.4 or
y = !2.1
!4 = 0 or z ! 7.5 = 0 or z ! 9.3 = 0
no solution
z = 7.5 or
z = 9.3
{0.4,!!2.1}
25.
{7.5,!9.3}
x 2 + 6x ! 27 = 0
26.
( x + 9 ) ( x ! 3) = 0
x + 9 = 0 or x ! 3 = 0
x = !9 or
27.
2x 2 + x ! 15 = 0
2x + 6x ! 5x ! 15 = 0
2x ( x + 3) ! 5 ( x + 3) = 0
( x + 3) ( 2x ! 5 ) = 0
x + 3 = 0 or 2x ! 5 = 0
x = !3 or 2x = 5
5
5
x = !3 or x = !!! "#!3,! %&
2 $
2'
2
x = 3!!!{!9,!3}
2x 2 + 5x = 3
28.
!11x = 3x 2 ! 4
2x 2 + 5x ! 3 = 0
3x 2 + 11x ! 4 = 0
2x 2 + 6x ! x ! 3 = 0
3x 2 + 12x ! x ! 4 = 0
2x ( x + 3) ! ( x + 3) = 0
3x ( x + 4 ) ! ( x + 4 ) = 0
( x + 3) ( 2x ! 1) = 0
( x + 4 ) ( 3x ! 1) = 0
x + 3 = 0 or 2x ! 1 = 0
x = !3 or 2x = 1
x + 4 = 0 or 3x ! 1 = 0
x = !4 or 3x = 1
1 "
1%
x = !3 or !!x = !!! #!3,! &
2 $
2'
307
1 "
1%
x = !4 or !x = !!! #!4,! &
3 $
3'
Chapter 4
Polynomials
10x 2 = 15x
29.
5x 2 = 7x
30.
10x 2 ! 15x = 0
5x 2 ! 7x = 0
5x ( 2x ! 3) = 0
x ( 5x ! 7 ) = 0
5x = 0 or 2x ! 3 = 0
x = 0 or
2x = 3
3 " 3%
x = !!!! #0,! &
2 $ 2'
x = 0 or
31.
x = 0 or 5x ! 7 = 0
x = 0 or
5x = 7
6 ( y ! 2 ) ! 3 ( y + 1) = 8
x = 0 or
32.
6y ! 12 ! 3y ! 3 = 8
3y ! 15 = 8
3y = 23
y=
4x + 3 ( x ! 9 ) = 6x + 1
4x + 3x ! 27 = 6x + 1
7x ! 27 = 6x + 1
7x ! 6x ! 27 = 6x ! 6x + 1
x ! 27 = 1
23 " 23 %
!!!! # &
3 $3'
!9 = y ( y + 6 )
33.
x = 28!!!!{28}
!62 = t ( t ! 16 ) + 2
34.
!9 = y + 6y
!62 = t 2 ! 16t + 2
2
y 2 + 6y + 9 = 0
t 2 ! 16t + 64 = 0
( y + 3)2 = 0
( t ! 8 )2 = 0
y+ 3= 0
t!8=0
y = !3!!!!{!3}
9 p 2 ! 15 p ! 6 = 0
35.
(
t = 8!!!!{8}
6y 2 + 2y = 48
6y 2 + 2y ! 48 = 0
36.
)
! 6 p + p ! 2) = 0
3 3p 2 ! 5 p ! 2 = 0
(
3 3p 2
(
2 3y
3 "# 3p ( p ! 2 ) + ( p ! 2 ) $% = 0
no solution
(
)
+ 9y ! 8y ! 24 ) = 0
2 3y 2 + y ! 24 = 0
2
2 "# 3y ( y + 3) ! 8 ( y + 3) $% = 0
3 ( p ! 2 ) ( 3p + 1) = 0
3 = 0 or
7 " 7%
x = !!!! #0,! &
5 $ 5'
2 ( y + 3) ( 3y ! 8 ) = 0
2 = 0 or
y + 3 = 0 or 3y ! 8 = 0
y = !3 or
3y = 8
8
no solution
y = !3 or
y=
3
8)
&
!!!!!!!!!!!!!!!!! '!3,! *
3+
(
p ! 2 = 0 or 3p + 1 = 0
p = 2 or
3p = !1
1
p = 2 or
p=!
3
1)
&
!!!!!!!!!!!!!!!!!! '2,!! *
3+
(
308
Section 4.8
37.
Solving Equations by Using the Zero Product Rule
( x + 1) ( 2x ! 1) ( x ! 3) = 0
38.
x + 1 = 0 or 2x ! 1 = 0 or x ! 3 = 0
x = !1 or
2x = 1 or
x=3
1
x = !1 or
x=
or
x=3
2
1 %
"
!!!!!!!!!! #!1,! ,!3&
2 '
$
39.
2x ( x ! 4 ) ( 4x + 3) = 0
2
2x = 0 or
x = 0 or
3
4
3%
"
!!!!!!!!!!!!!!! #0,!4,!! &
4'
$
x = 0 or
( y ! 3) ( y + 4 ) = 8
40.
x = 4 or
t 2 + 15t + 50 = 6
y 2 + y ! 20 = 0
t 2 + 15t + 44 = 0
(t + 11) (t + 4 ) = 0
t + 11 = 0 or t + 4 = 0
y + 5 = 0 or y ! 4 = 0
y = !5 or
y = 4!!!!{!5,!4}
t = !11 or
( 2a ! 1) ( a ! 1) = 6
2a ! 3a + 1 = 6
6w 2 + w = 2
2a 2 ! 3a ! 5 = 0
6w 2 + w ! 2 = 0
( 2a ! 5 ) ( a + 1) = 0
( 3w + 2 ) ( 2w ! 1) = 0
2a ! 5 = 0 or a + 1 = 0
2a = 5 or
a = !1
5
or
2
3w + 2 = 0 or 2w ! 1 = 0
3w = !2 or 2w = 1
"5
%
a = !1!!!! # ,!!1&
$2
'
w=!
p 2 + ( p + 7 ) = 169
43.
x 2 + x 2 + 4x + 4 = 100
2 p 2 + 14 p ! 120 = 0
2x 2 + 4x ! 96 = 0
)
(
)
2 p 2 + 7 p ! 60 = 0
2 x 2 + 2x ! 48 = 0
2 ( p + 12 ) ( p ! 5 ) = 0
2 ( x + 8)( x ! 6) = 0
p + 12 = 0 or p ! 5 = 0
2 " 0 or
x + 8 = 0 or x ! 6 = 0
p = !12 or p = 5!!!{!12,!5}
3t ( t + 5 ) ! t 2 = 2t 2 + 4t ! 1
x = !8 or
46.
3t 2 + 15t ! t 2 = 2t 2 + 4t ! 1
11t = !1
t=!
a 2 ! 4a ! 2 = ( a + 3) ( a ! 5 )
a 2 ! 4a ! 2 = a 2 ! 2a ! 15
!2a = !13
1 " 1%
!!!! #! &
11 $ 11 '
a=
309
1 " 2 1%
w = !! #! ,! &
2 $ 3 2'
2
p 2 + p 2 + 14 p + 49 = 169
2 " 0 or
45.
2
or
3
x 2 + ( x + 2 ) = 100
44.
2
(
t = !4!!!!{!11,!!4 }
w ( 6w + 1) = 2
42.
2
a=
x=!
(t + 10 ) (t + 5 ) = 6
y 2 + y ! 12 = 8
( y + 5)( y ! 4 ) = 0
41.
x ! 4 = 0 or 4x + 3 = 0
x = 4 or
4x = !3
13 "13 %
!!!! # &
2 $2'
x = 6!!!!{!8,!6}
Chapter 4
47.
Polynomials
2x 3 ! 8x 2 ! 24x = 0
(
48.
)
(
)
2x x 2 ! 4x ! 12 = 0
2 p p 2 + 10 p + 21 = 0
2x ( x ! 6 ) ( x + 2 ) = 0
2 p ( p + 7 ) ( p + 3) = 0
2x = 0 or x ! 6 = 0 or x + 2 = 0
!!x = 0 or
2 p = 0 or
x = 6 or !!x = !2!!!{0,!6,!!2}
w 3 = 16w
49.
(
50.
3x ( 2x + 3) ( 2x ! 3) = 0
!!3x = 0 or 2x + 3 = 0 or 2x ! 3 = 0
!!!!x = 0 or
2x = !3 or
2x = 3
3
3 "
3 3%
!!!!x = 0 or !!x = ! or!! x = !! #0.!! ,! &
2
2 $
2 2'
w + 4 = 0 or w ! 4 = 0
x = !4 or !x = 4
{0,!!4,!4}
0 = 2x 3 + 5x 2 ! 18x ! 45
0 = x 2 ( 2x + 5 ) ! 9 ( 2x + 5 )
(
)
3x 4x 2 ! 9 = 0
w (w + 4 )(w ! 4 ) = 0
0 = ( 2x + 5 ) x 2 ! 9
12x 3 = 27x
12x 3 ! 27x = 0
(
)
w w 2 ! 16 = 0
w = 0 or
w = 0 or
p + 7 = 0 or p + 3 = 0
!! p = 0 or !!!!!p = !7 or !p = !3!!!{0,!!7,!!3}
w 3 ! 16w = 0
51.
2 p 3 + 20 p 2 + 42 p = 0
52.
)
(
0 = ( 3y + 1) y 2 ! 16
0 = ( 2x + 5 ) ( x + 3) ( x ! 3)
2x + 5 = 0 or x + 3 = 0 or x ! 3 = 0
2x = !5 or
x = !3 or
x=3
5
x = ! or
x = !3 or
x=3
2
" 5
%
!!!!!! #! ,!!3,!3&
2
$
'
53. Let x = the number
x 2 + 5 = 30
0 = 3y 3 + y 2 ! 48y ! 16
0 = y 2 ( 3y + 1) ! 16 ( 3y + 1)
0 = ( 3y + 1) ( y + 4 ) ( y ! 4 )
3y + 1 = 0
or y + 4 = 0 or y ! 4 = 0
3y = !1 or
y = !4 or y = 4
1
y = ! or
y = !4 or
y=4
3
" 1
%
!!!!!! #! ,!!4,!4 &
3
$
'
54.
Let x = the number
x 2 ! 4 = 77
x 2 ! 25 = 0
x 2 ! 81 = 0
( x + 5)( x ! 5) = 0
( x + 9)( x ! 9) = 0
x + 5 = 0 or x ! 5 = 0
x = !5 or x = 5
55. Let x = the number
x 2 = x + 12
)
x + 9 = 0 or x ! 9 = 0
x = !9 or x = 9
56.
Let x = the number
x 2 = x + 20
x 2 ! x ! 12 = 0
x 2 ! x ! 20 = 0
( x + 3) ( x ! 4 ) = 0
( x + 4 )( x ! 5) = 0
x + 4 = 0 or x ! 5 = 0
x = ! 4 or x = 5
x + 3 = 0 or x ! 4 = 0
x = !3 or
x=4
310