Precalculus 2e

# Chapter 3

### 3.1Complex Numbers

1 .

$−24 =0+2i 6 −24 =0+2i 6$

2 .
3 .

$(3−4i)−(2+5i)=1−9i (3−4i)−(2+5i)=1−9i$

4 .

$−8−24i −8−24i$

5 .

$18+i 18+i$

6 .

$102−29i 102−29i$

7 .

$− 3 17 + 5i 17 − 3 17 + 5i 17$

1 .

The path passes through the origin and has vertex at $( −4,7 ), ( −4,7 ),$ so $(h)x=– 7 16 (x+4) 2 +7. (h)x=– 7 16 (x+4) 2 +7.$ To make the shot, $h( −7.5 ) h( −7.5 )$ would need to be about 4 but $h(–7.5)≈1.64; h(–7.5)≈1.64;$ he doesn’t make it.

2 .

$g(x)= x 2 −6x+13 g(x)= x 2 −6x+13$ in general form; $g(x)= (x−3) 2 +4 g(x)= (x−3) 2 +4$ in standard form

3 .

The domain is all real numbers. The range is $f(x)≥ 8 11 , f(x)≥ 8 11 ,$ or $[ 8 11 ,∞ ). [ 8 11 ,∞ ).$

4 .

y-intercept at (0, 13), No $x- x-$ intercepts

5 .
1. 3 seconds
2. 256 feet
3. 7 seconds

### 3.3Power Functions and Polynomial Functions

1 .

$f(x) f(x)$ is a power function because it can be written as $f(x)=8 x 5 . f(x)=8 x 5 .$ The other functions are not power functions.

2 .

As $x x$ approaches positive or negative infinity, $f( x ) f( x )$ decreases without bound: as $x→±∞ x→±∞$, $f(x)→−∞ f(x)→−∞$ because of the negative coefficient.

3 .

The degree is 6. The leading term is $− x 6 . − x 6 .$ The leading coefficient is $−1. −1.$

4 .

As $x→∞,f(x)→−∞;asx→−∞,f(x)→−∞. x→∞,f(x)→−∞;asx→−∞,f(x)→−∞.$ It has the shape of an even degree power function with a negative coefficient.

5 .

The leading term is $0.2 x 3 , 0.2 x 3 ,$ so it is a degree 3 polynomial. As $x x$ approaches positive infinity, $f( x ) f( x )$ increases without bound; as $x x$ approaches negative infinity, $f( x ) f( x )$ decreases without bound.

6 .

y-intercept $(0,0); (0,0);$ x-intercepts $(0,0),(–2,0), (0,0),(–2,0),$ and $(5,0) (5,0)$

7 .

There are at most 12 $x- x-$ intercepts and at most 11 turning points.

8 .

The end behavior indicates an odd-degree polynomial function; there are 3 $x- x-$ intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

9 .

The $x- x-$ intercepts are $(2,0),(−1,0), (2,0),(−1,0),$ and $(5,0), (5,0),$ the y-intercept is $(0,2), (0,2),$ and the graph has at most 2 turning points.

### 3.4Graphs of Polynomial Functions

1 .

y-intercept $(0,0); (0,0);$ x-intercepts $(0,0),(–5,0),(2,0), (0,0),(–5,0),(2,0),$ and $(3,0) (3,0)$

2 .

The graph has a zero of –5 with multiplicity 3, a zero of –1 with multiplicity 2, and a zero of 3 with multiplicity 2.

3 .
4 .

Because $f f$ is a polynomial function and since $f(1) f(1)$ is negative and $f(2) f(2)$ is positive, there is at least one real zero between $x=1 x=1$ and $x=2. x=2.$

5 .

$f(x)=− 1 8 (x−2) 3 (x+1) 2 (x−4) f(x)=− 1 8 (x−2) 3 (x+1) 2 (x−4)$

6 .

The minimum occurs at approximately the point $(0,−6.5), (0,−6.5),$ and the maximum occurs at approximately the point $(3.5,7). (3.5,7).$

### 3.5Dividing Polynomials

1 .

$4 x 2 −8x+15− 78 4x+5 4 x 2 −8x+15− 78 4x+5$

2 .

$3 x 3 −3 x 2 +21x−150+ 1,090 x+7 3 x 3 −3 x 2 +21x−150+ 1,090 x+7$

3 .

$3 x 2 −4x+1 3 x 2 −4x+1$

### 3.6Zeros of Polynomial Functions

1 .

$f(−3)=−412 f(−3)=−412$

2 .

The zeros are 2, –2, and –4.

3 .

There are no rational zeros.

4 .

The zeros are

5 .

$f(x)=− 1 2 x 3 + 5 2 x 2 −2x+10 f(x)=− 1 2 x 3 + 5 2 x 2 −2x+10$

6 .

There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.

7 .

3 meters by 4 meters by 7 meters

### 3.7Rational Functions

1 .

End behavior: as $x→±∞,f(x)→0; x→±∞,f(x)→0;$ Local behavior: as $x→0,f(x)→∞ x→0,f(x)→∞$ (there are no x- or y-intercepts)

2 .

The function and the asymptotes are shifted 3 units right and 4 units down. As $x→3,f(x)→∞, x→3,f(x)→∞,$ and as $x→±∞,f(x)→−4. x→±∞,f(x)→−4.$

The function is $f(x)= 1 (x−3) 2 −4. f(x)= 1 (x−3) 2 −4.$

3 .

$12 11 12 11$

4 .

The domain is all real numbers except $x=1 x=1$ and $x=5. x=5.$

5 .

Removable discontinuity at $x=5. x=5.$ Vertical asymptotes: $x=0,x=1. x=0,x=1.$

6 .

Vertical asymptotes at $x=2 x=2$ and $x=–3; x=–3;$ horizontal asymptote at $y=4. y=4.$

7 .

For the transformed reciprocal squared function, we find the rational form. $f(x)= 1 (x−3) 2 −4= 1−4 (x−3) 2 (x−3) 2 = 1−4( x 2 −6x+9) (x−3)(x−3) = −4 x 2 +24x−35 x 2 −6x+9 f(x)= 1 (x−3) 2 −4= 1−4 (x−3) 2 (x−3) 2 = 1−4( x 2 −6x+9) (x−3)(x−3) = −4 x 2 +24x−35 x 2 −6x+9$

Because the numerator is the same degree as the denominator we know that as $x→±∞,f(x)→−4;soy=–4 x→±∞,f(x)→−4;soy=–4$ is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is $x=3, x=3,$ because as $x→3,f(x)→∞. x→3,f(x)→∞.$ We then set the numerator equal to 0 and find the x-intercepts are at $(2.5,0) (2.5,0)$ and $(3.5,0). (3.5,0).$ Finally, we evaluate the function at 0 and find the y-intercept to be at $( 0, −35 9 ). ( 0, −35 9 ).$

8 .

Horizontal asymptote at $y= 1 2 . y= 1 2 .$ Vertical asymptotes at $x=1andx=3. x=1andx=3.$ y-intercept at $( 0, 4 3 . ) ( 0, 4 3 . )$

x-intercepts at $(–2,0) (–2,0)$ is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. $(2,0) (2,0)$ is a single zero and the graph crosses the axis at this point.

1 .

$f −1 ( f( x ) )= f −1 ( x+5 3 )=3( x+5 3 )−5=( x−5 )+5=x f −1 ( f( x ) )= f −1 ( x+5 3 )=3( x+5 3 )−5=( x−5 )+5=x$ and $f( f −1 ( x ) )=f( 3x−5 )= ( 3x−5 )+5 3 = 3x 3 =x f( f −1 ( x ) )=f( 3x−5 )= ( 3x−5 )+5 3 = 3x 3 =x$

2 .

$f −1 (x)= x 3 −4 f −1 (x)= x 3 −4$

3 .

$f −1 (x)= x−1 f −1 (x)= x−1$

4 .

$f −1 (x)= x 2 −3 2 ,x≥0 f −1 (x)= x 2 −3 2 ,x≥0$

5 .

$f −1 (x)= 2x+3 x−1 f −1 (x)= 2x+3 x−1$

### 3.9Modeling Using Variation

1 .

$128 3 128 3$

2 .

$9 2 9 2$

3 .

$x=20 x=20$

### 3.1 Section Exercises

1 .

Add the real parts together and the imaginary parts together.

3 .

$i i$ times $i i$ equals –1, which is not imaginary. (answers vary)

5 .

$−8+2i −8+2i$

7 .

$14+7i 14+7i$

9 .

$− 23 29 + 15 29 i − 23 29 + 15 29 i$

11 .

2 real and 0 nonreal

13 .
15 .
17 .

$8−i 8−i$

19 .

$−11+4i −11+4i$

21 .

$2−5i 2−5i$

23 .

$6+15i 6+15i$

25 .

$−16+32i −16+32i$

27 .

$−4−7i −4−7i$

29 .

25

31 .

$2− 2 3 i 2− 2 3 i$

33 .

$4−6i 4−6i$

35 .

$2 5 + 11 5 i 2 5 + 11 5 i$

37 .

$15i 15i$

39 .

$1+i 3 1+i 3$

41 .

$1 1$

43 .

$−1 −1$

45 .

128i

47 .

$( 3 2 + 1 2 i ) 6 =−1 ( 3 2 + 1 2 i ) 6 =−1$

49 .

$3i 3i$

51 .

0

53 .

5 – 5i

55 .

$−2i −2i$

57 .

$9 2 − 9 2 i 9 2 − 9 2 i$

### 3.2 Section Exercises

1 .

When written in that form, the vertex can be easily identified.

3 .

If $a=0 a=0$ then the function becomes a linear function.

5 .

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

7 .

$g(x)= (x+1) 2 −4, g(x)= (x+1) 2 −4,$ Vertex $( −1,−4 ) ( −1,−4 )$

9 .

$f(x)= ( x+ 5 2 ) 2 − 33 4 , f(x)= ( x+ 5 2 ) 2 − 33 4 ,$ Vertex $( − 5 2 ,− 33 4 ) ( − 5 2 ,− 33 4 )$

11 .

$f(x)=3 (x−1) 2 −12, f(x)=3 (x−1) 2 −12,$ Vertex $(1,−12) (1,−12)$

13 .

$f(x)=3 ( x− 5 6 ) 2 − 37 12 , f(x)=3 ( x− 5 6 ) 2 − 37 12 ,$ Vertex $( 5 6 ,− 37 12 ) ( 5 6 ,− 37 12 )$

15 .

Minimum is $− 17 2 − 17 2$ and occurs at $5 2 . 5 2 .$ Axis of symmetry is $x= 5 2 . x= 5 2 .$

17 .

Minimum is $− 17 16 − 17 16$ and occurs at $− 1 8 . − 1 8 .$ Axis of symmetry is $x=− 1 8 . x=− 1 8 .$

19 .

Minimum is $− 7 2 − 7 2$ and occurs at $−3. −3.$ Axis of symmetry is $x=−3. x=−3.$

21 .

Domain is $( −∞,∞ ). ( −∞,∞ ).$ Range is $[2,∞). [2,∞).$

23 .

Domain is $( −∞,∞ ). ( −∞,∞ ).$ Range is $[−5,∞). [−5,∞).$

25 .

Domain is $( −∞,∞ ). ( −∞,∞ ).$ Range is $[−12,∞). [−12,∞).$

27 .

${ 2i 2 ,−2i 2 } { 2i 2 ,−2i 2 }$

29 .

${ 3i 3 ,−3i 3 } { 3i 3 ,−3i 3 }$

31 .

${2+i,2−i} {2+i,2−i}$

33 .

${2+3i,2−3i} {2+3i,2−3i}$

35 .

${5+i,5−i} {5+i,5−i}$

37 .

${2+2 6 ,2−2 6 } {2+2 6 ,2−2 6 }$

39 .

${ − 1 2 + 3 2 i,− 1 2 − 3 2 i } { − 1 2 + 3 2 i,− 1 2 − 3 2 i }$

41 .

${ − 3 5 + 1 5 i,− 3 5 − 1 5 i } { − 3 5 + 1 5 i,− 3 5 − 1 5 i }$

43 .

${ − 1 2 + 1 2 i 7 ,− 1 2 − 1 2 i 7 } { − 1 2 + 1 2 i 7 ,− 1 2 − 1 2 i 7 }$

45 .

$f(x)= x 2 −4x+4 f(x)= x 2 −4x+4$

47 .

$f(x)= x 2 +1 f(x)= x 2 +1$

49 .

$f(x)= 6 49 x 2 + 60 49 x+ 297 49 f(x)= 6 49 x 2 + 60 49 x+ 297 49$

51 .

$f(x)=− x 2 +1 f(x)=− x 2 +1$

53 .

Vertex $( 1,−1 ), ( 1,−1 ),$ Axis of symmetry is $x=1. x=1.$ Intercepts are $(0,0),(2,0). (0,0),(2,0).$

55 .

Vertex $( 5 2 , −49 4 ), ( 5 2 , −49 4 ),$ Axis of symmetry is $x= 5 2 , x= 5 2 ,$ intercepts: $(6,0), (−1,0). (6,0), (−1,0).$

57 .

Vertex $( 5 4 ,− 39 8 ), ( 5 4 ,− 39 8 ),$ Axis of symmetry is $x= 5 4 . x= 5 4 .$ Intercepts are $( 0,−8 ). ( 0,−8 ).$

59 .

$f(x)= x 2 −4x+1 f(x)= x 2 −4x+1$

61 .

$f(x)=−2 x 2 +8x−1 f(x)=−2 x 2 +8x−1$

63 .

$f(x)= 1 2 x 2 −3x+ 7 2 f(x)= 1 2 x 2 −3x+ 7 2$

65 .

$f(x)= x 2 +1 f(x)= x 2 +1$

67 .

$f(x)=2− x 2 f(x)=2− x 2$

69 .

$f(x)=2 x 2 f(x)=2 x 2$

71 .

The graph is shifted up or down (a vertical shift).

73 .

50 feet

75 .

Domain is $(−∞,∞). (−∞,∞).$ Range is $[−2,∞). [−2,∞).$

77 .

Domain is $(−∞,∞) (−∞,∞)$ Range is $(−∞,11]. (−∞,11].$

79 .

$f(x)=2 x 2 −1 f(x)=2 x 2 −1$

81 .

$f(x)=3 x 2 −9 f(x)=3 x 2 −9$

83 .

$f(x)=5 x 2 −77 f(x)=5 x 2 −77$

85 .

50 feet by 50 feet. Maximize $f(x)=− x 2 +100x. f(x)=− x 2 +100x.$

87 .

125 feet by 62.5 feet. Maximize $f(x)=−2 x 2 +250x. f(x)=−2 x 2 +250x.$

89 .

$6 6$ and $−6; −6;$ product is –36; maximize $f(x)= x 2 +12x. f(x)= x 2 +12x.$

91 .

2909.56 meters

93 .

\$10.70

### 3.3 Section Exercises

1 .

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

3 .

As $x x$ decreases without bound, so does $f( x ). f( x ).$ As $x x$ increases without bound, so does $f( x ). f( x ).$

5 .

The polynomial function is of even degree and leading coefficient is negative.

7 .

$f(x)f(x)$ is a power function because it contains a variable base raised to a fixed power. It is also a polynomial, with all coefficients except one equal to zero.

9 .

Neither

11 .

Neither

13 .

Degree = 2, Coefficient = –2

15 .

Degree =4, Coefficient = –2

17 .

As $x→∞ x→∞$, $f(x)→∞f(x)→∞$, as $x→−∞x→−∞$, $f(x)→∞ f(x)→∞$

19 .

As $x→−∞ x→−∞$, $f(x)→−∞f(x)→−∞$, as $x→∞x→∞$, $f(x)→−∞ f(x)→−∞$

21 .

As $x→−∞ x→−∞$, $f(x)→−∞f(x)→−∞$,as $x→∞x→∞$, $f(x)→−∞ f(x)→−∞$

23 .

As $x→∞x→∞$, $f(x)→∞f(x)→∞$, as $x→−∞x→−∞$,$f(x)→−∞ f(x)→−∞$

25 .

y-intercept is $(0,12), (0,12),$ t-intercepts are

27 .

y-intercept is $(0,−16). (0,−16).$ x-intercepts are $(2,0) (2,0)$ and $(−2,0). (−2,0).$

29 .

y-intercept is $(0,0). (0,0).$ x-intercepts are $(0,0),(4,0), (0,0),(4,0),$ and $( −2,0 ). ( −2,0 ).$

31 .

3

33 .

5

35 .

3

37 .

5

39 .

Yes. Number of turning points is 2. Least possible degree is 3.

41 .

Yes. Number of turning points is 1. Least possible degree is 2.

43 .

Yes. Number of turning points is 0. Least possible degree is 1.

44 .

No.

45 .

Yes. Number of turning points is 0. Least possible degree is 1.

47 .
$x x$ $f( x ) f( x )$
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

as $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

49 .
$x x$ $f( x ) f( x )$
10 –504
100 –941,094
–10 1,716
–100 1,061,106

as $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→−∞ f(x)→−∞$

51 .

The $y- y-$ intercept is $( 0,0 ). ( 0,0 ).$ The $x- x-$ intercepts are $( 0,0 ),( 2,0 ). ( 0,0 ),( 2,0 ).$As $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

53 .

The $y- y-$ intercept is $( 0,0 ) ( 0,0 )$ . The $x- x-$ intercepts are $( 0,0 ),( 5,0 ),( 7,0 ). ( 0,0 ),( 5,0 ),( 7,0 ).$ As $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

55 .

The $y- y-$ intercept is $( 0,0 ). ( 0,0 ).$ The $x- x-$ intercept is $( −4,0 ),( 0,0 ),( 4,0 ). ( −4,0 ),( 0,0 ),( 4,0 ).$ As $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

57 .

The $y- y-$ intercept is $( 0,−81 ). ( 0,−81 ).$ The $x- x-$ intercept are $( 3,0 ),( −3,0 ). ( 3,0 ),( −3,0 ).$ As $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

59 .

The $y- y-$ intercept is $( 0,0 ). ( 0,0 ).$ The $x- x-$ intercepts are $( −3,0 ),( 0,0 ),( 5,0 ). ( −3,0 ),( 0,0 ),( 5,0 ).$ As $x→−∞ x→−∞$, $f(x)→∞f(x)→∞$, as $x→∞x→∞$, $f(x)→∞ f(x)→∞$

61 .

$f(x)= x 2 −4 f(x)= x 2 −4$

63 .

$f(x)= x 3 −4 x 2 +4x f(x)= x 3 −4 x 2 +4x$

65 .

$f(x)= x 4 +1 f(x)= x 4 +1$

67 .

$V(m)=8 m 3 +36 m 2 +54m+27 V(m)=8 m 3 +36 m 2 +54m+27$

69 .

$V(x)=4 x 3 −32 x 2 +64x V(x)=4 x 3 −32 x 2 +64x$

### 3.4 Section Exercises

1 .

The $x- x-$ intercept is where the graph of the function crosses the $x- x-$ axis, and the zero of the function is the input value for which $f(x)=0. f(x)=0.$

3 .

If we evaluate the function at $a a$ and at $b b$ and the sign of the function value changes, then we know a zero exists between $a a$ and $b. b.$

5 .

There will be a factor raised to an even power.

7 .

$(−2,0),(3,0),(−5,0) (−2,0),(3,0),(−5,0)$

9 .

$(3,0),(−1,0),(0,0) (3,0),(−1,0),(0,0)$

11 .

$( 0,0 ),( −5,0 ),( 2,0 ) ( 0,0 ),( −5,0 ),( 2,0 )$

13 .

$( 0,0 ),( −5,0 ),( 4,0 ) ( 0,0 ),( −5,0 ),( 4,0 )$

15 .

$( 2,0 ),( −2,0 ),( −1,0 ) ( 2,0 ),( −2,0 ),( −1,0 )$

17 .

$(−2,0),(2,0),( 1 2 ,0 ) (−2,0),(2,0),( 1 2 ,0 )$

19 .

$( 1,0 ),( −1,0 ) ( 1,0 ),( −1,0 )$

21 .

$(0,0),( 3 ,0),(− 3 ,0) (0,0),( 3 ,0),(− 3 ,0)$

23 .

$( 0,0 ) ( 0,0 )$, $( 1,0 )( 1,0 )$, $( −1,0 )( −1,0 )$, $( 2,0 )( 2,0 )$, $( −2,0 ) ( −2,0 )$

25 .

$f( 2 )=–10 f( 2 )=–10$ and $f( 4 )=28. f( 4 )=28.$ Sign change confirms.

27 .

$f( 1 )=3 f( 1 )=3$ and $f( 3 )=–77. f( 3 )=–77.$ Sign change confirms.

29 .

$f( 0.01 )=1.000001 f( 0.01 )=1.000001$ and $f( 0.1 )=–7.999. f( 0.1 )=–7.999.$ Sign change confirms.

31 .

0 with multiplicity 2, $− 3 2 − 3 2$ with multiplicity 5, 4 with multiplicity 2

33 .

0 with multiplicity 2, –2 with multiplicity 2

35 .

$− 2 3 − 2 3$ with multiplicity $5,55,5$ with multiplicity $2 2$

37 .

$0 0$ with multiplicity $4,24,2$ with multiplicity $1,–11,–1$ with multiplicity $1 1$

39 .

$3 2 3 2$ with multiplicity 2, 0 with multiplicity 3

41 .

$0 0$ with multiplicity $6, 2 3 6, 2 3$ with multiplicity $2 2$

43 .

x-intercepts, $( 1, 0 ) ( 1, 0 )$ with multiplicity 2, $( –4, 0 ) ( –4, 0 )$ with multiplicity 1, $y- y-$ intercept $( 0, 4 ) ( 0, 4 )$ . As $x→−∞x→−∞$, $g(x)→−∞g(x)→−∞$, as $x→∞x→∞$, $g(x)→∞g(x)→∞$.

45 .

x-intercepts $(3,0) (3,0)$ with multiplicity 3, $(2,0) (2,0)$ with multiplicity 2, $y- y-$ intercept $(0,–108) (0,–108)$. As $x→−∞x→−∞$, $k(x)→−∞k(x)→−∞$, as $x→∞ x→∞$, $k(x)→∞. k(x)→∞.$

47 .

x-intercepts $(0, 0 ), (–2, 0), (4, 0) (0, 0 ), (–2, 0), (4, 0)$ with multiplicity 1, $y y$-intercept $(0, 0). (0, 0).$ As $x→−∞ x→−∞$, $n(x)→∞n(x)→∞$, as $x→∞x→∞$, $n(x)→−∞. n(x)→−∞.$

49 .

$f(x)=− 2 9 (x−3)(x+1)(x+3) f(x)=− 2 9 (x−3)(x+1)(x+3)$

51 .

$f(x)= 1 4 (x+2) 2 (x−3) f(x)= 1 4 (x+2) 2 (x−3)$

53 .

–4, –2, 1, 3 with multiplicity 1

55 .

–2, 3 each with multiplicity 2

57 .

$f(x)=− 2 3 (x+2)(x−1)(x−3) f(x)=− 2 3 (x+2)(x−1)(x−3)$

59 .

$f(x)= 1 3 (x−3) 2 (x−1) 2 (x+3) f(x)= 1 3 (x−3) 2 (x−1) 2 (x+3)$

61 .

$f(x)=−15 (x−1) 2 (x−3) 3 f(x)=−15 (x−1) 2 (x−3) 3$

63 .

$f(x)=−2( x+3 )( x+2 )( x−1 ) f(x)=−2( x+3 )( x+2 )( x−1 )$

65 .

$f(x)=− 3 2 ( 2x−1 ) 2 ( x−6 )( x+2 ) f(x)=− 3 2 ( 2x−1 ) 2 ( x−6 )( x+2 )$

67 .

local max local min

69 .

global min

71 .

global min

73 .

$f(x)= (x−500) 2 (x+200) f(x)= (x−500) 2 (x+200)$

75 .

$f(x)=4 x 3 −36 x 2 +80x f(x)=4 x 3 −36 x 2 +80x$

77 .

$f(x)=4 x 3 −36 x 2 +60x+100 f(x)=4 x 3 −36 x 2 +60x+100$

79 .

$f(x)=9π( x 3 +5 x 2 +8x+4) f(x)=9π( x 3 +5 x 2 +8x+4)$

### 3.5 Section Exercises

1 .

The binomial is a factor of the polynomial.

3 .

$x+6+ 5 x-1 x+6+ 5 x-1$, quotient: $x+6 x+6$, remainder: $5 5$

5 .

$3x+2 3x+2$, quotient:  $3x+2 3x+2$, remainder:  $0 0$

7 .

$x−5 x−5$, quotient: $x−5 x−5$, remainder: $0 0$

9 .

$2x−7+ 16 x+2 2x−7+ 16 x+2$, quotient: $2x−7 2x−7$, remainder: $16 16$

11 .

$x−2+ 6 3x+1 x−2+ 6 3x+1$, quotient: $x−2 x−2$, remainder: $6 6$

13 .

$2 x 2 −3x+5 2 x 2 −3x+5$, quotient: $2 x 2 −3x+5 2 x 2 −3x+5$, remainder: $0 0$

15 .

$2 x 2 +2x+1+ 10 x−4 2 x 2 +2x+1+ 10 x−4$

17 .

$2 x 2 −7x+1− 2 2x+1 2 x 2 −7x+1− 2 2x+1$

19 .

$3 x 2 −11x+34− 106 x+3 3 x 2 −11x+34− 106 x+3$

21 .

$x 2 +5x+1 x 2 +5x+1$

23 .

$4 x 2 −21x+84− 323 x+4 4 x 2 −21x+84− 323 x+4$

25 .

$x 2 −14x+49 x 2 −14x+49$

27 .

$3 x 2 +x+ 2 3x−1 3 x 2 +x+ 2 3x−1$

29 .

$x 3 −3x+1 x 3 −3x+1$

31 .

$x 3 − x 2 +2 x 3 − x 2 +2$

33 .

$x 3 −6 x 2 +12x−8 x 3 −6 x 2 +12x−8$

35 .

$x 3 −9 x 2 +27x−27 x 3 −9 x 2 +27x−27$

37 .

$2 x 3 −2x+2 2 x 3 −2x+2$

39 .

Yes $( x−2 )(3 x 3 −5) ( x−2 )(3 x 3 −5)$

41 .

Yes $( x−2 )(4 x 3 +8 x 2 +x+2) ( x−2 )(4 x 3 +8 x 2 +x+2)$

43 .

No

45 .

$(x−1)( x 2 +2x+4) (x−1)( x 2 +2x+4)$

47 .

$(x−5)( x 2 +x+1) (x−5)( x 2 +x+1)$

49 .

Quotient: $4 x 2 +8x+164 x 2 +8x+16$, remainder: $−1 −1$

51 .

Quotient: $3 x 2 +3x+53 x 2 +3x+5$, remainder: $0 0$

53 .

Quotient: $x 3 −2 x 2 +4x−8 x 3 −2 x 2 +4x−8$, remainder: $−6 −6$

55 .

$x 6 − x 5 + x 4 − x 3 + x 2 −x+1 x 6 − x 5 + x 4 − x 3 + x 2 −x+1$

57 .

$x 3 − x 2 +x−1+ 1 x+1 x 3 − x 2 +x−1+ 1 x+1$

59 .

$1+ 1+i x−i 1+ 1+i x−i$

61 .

$1+ 1−i x+i 1+ 1−i x+i$

63 .

$x 2 −ix−1+ 1−i x−i x 2 −ix−1+ 1−i x−i$

65 .

$2 x 2 +3 2 x 2 +3$

67 .

$2x+3 2x+3$

69 .

$x+2 x+2$

71 .

$x−3 x−3$

73 .

$3 x 2 −2 3 x 2 −2$

### 3.6 Section Exercises

1 .

The theorem can be used to evaluate a polynomial.

3 .

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

5 .

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

7 .

$−106 −106$

9 .

$0 0$

11 .

$255 255$

13 .

$−1 −1$

15 .

$−2,1, 1 2 −2,1, 1 2$

17 .

$−2 −2$

19 .

$−3 −3$

21 .

$− 5 2 , 6 ,− 6 − 5 2 , 6 ,− 6$

23 .

$2,−4,− 3 2 2,−4,− 3 2$

25 .

$4,−4,−5 4,−4,−5$

27 .

$5,−3,− 1 2 5,−3,− 1 2$

29 .

$1 2 , 1+ 5 2 , 1− 5 2 1 2 , 1+ 5 2 , 1− 5 2$

31 .

$3 2 3 2$

33 .

$2,3,−1,−2 2,3,−1,−2$

35 .

$1 2 ,− 1 2 ,2,−3 1 2 ,− 1 2 ,2,−3$

37 .

$−1,−1, 5 ,− 5 −1,−1, 5 ,− 5$

39 .

$− 3 4 ,− 1 2 − 3 4 ,− 1 2$

41 .

$2,3+2i,3−2i 2,3+2i,3−2i$

43 .

$− 2 3 ,1+2i,1−2i − 2 3 ,1+2i,1−2i$

45 .

$− 1 2 ,1+4i,1−4i − 1 2 ,1+4i,1−4i$

47 .

1 positive, 1 negative

49 .

3 or 1 positive, 0 negative

51 .

0 positive, 3 or 1 negative

53 .

2 or 0 positive, 2 or 0 negative

55 .

2 or 0 positive, 2 or 0 negative

57 .

$±5,±1,± 5 2 ±5,±1,± 5 2$

59 .

$±1,± 1 2 ,± 1 3 ,± 1 6 ±1,± 1 2 ,± 1 3 ,± 1 6$

61 .

$1, 1 2 ,− 1 3 1, 1 2 ,− 1 3$

63 .

$2, 1 4 ,− 3 2 2, 1 4 ,− 3 2$

65 .

$5 4 5 4$

67 .

$f(x)= 4 9 ( x 3 + x 2 −x−1 ) f(x)= 4 9 ( x 3 + x 2 −x−1 )$

69 .

$f(x)=− 1 5 ( 4 x 3 −x ) f(x)=− 1 5 ( 4 x 3 −x )$

71 .

8 by 4 by 6 inches

73 .

5.5 by 4.5 by 3.5 inches

75 .

8 by 5 by 3 inches

77 .

Radius = 6 meters, Height = 2 meters

79 .

Radius = 2.5 meters, Height = 4.5 meters

### 3.7 Section Exercises

1 .

The rational function will be represented by a quotient of polynomial functions.

3 .

The numerator and denominator must have a common factor.

5 .

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

7 .

9 .

11 .

V.A. at $x=– 2 5 ; x=– 2 5 ;$ H.A. at $y=0; y=0;$ Domain is all reals $x≠– 2 5 x≠– 2 5$

13 .

V.A. at $x=4,–9; x=4,–9;$ H.A. at $y=0; y=0;$ Domain is all reals $x≠4,–9 x≠4,–9$

15 .

V.A. at $x=0,4,−4; x=0,4,−4;$ H.A. at $y=0; y=0;$ Domain is all reals $x≠0,4,–4 x≠0,4,–4$

17 .

V.A. at $x=5; x=5;$ H.A. at $y=0; y=0;$ Domain is all reals $x≠5,−5 x≠5,−5$

19 .

V.A. at $x= 1 3 ; x= 1 3 ;$ H.A. at $y=− 2 3 ; y=− 2 3 ;$ Domain is all reals $x≠ 1 3 . x≠ 1 3 .$

21 .

none

23 .

25 .

Local behavior: $x→− 1 2 + ,f(x)→−∞,x→− 1 2 − ,f(x)→∞ x→− 1 2 + ,f(x)→−∞,x→− 1 2 − ,f(x)→∞$

End behavior: $x→±∞,f(x)→ 1 2 x→±∞,f(x)→ 1 2$

27 .

Local behavior: $x→ 6 + ,f(x)→−∞,x→ 6 − ,f(x)→∞, x→ 6 + ,f(x)→−∞,x→ 6 − ,f(x)→∞,$ End behavior: $x→±∞,f(x)→−2 x→±∞,f(x)→−2$

29 .

Local behavior: $x→ 1 3 + ,f(x)→∞,x→ 1 3 − , x→ 1 3 + ,f(x)→∞,x→ 1 3 − ,$ $f(x)→−∞,x→− 5 2 − ,f(x)→∞,x→− 5 2 + f(x)→−∞,x→− 5 2 − ,f(x)→∞,x→− 5 2 +$, $f(x)→−∞ f(x)→−∞$

End behavior: $x→±∞, x→±∞,$ $f(x)→ 1 3 f(x)→ 1 3$

31 .

$y=2x+4 y=2x+4$

33 .

$y=2x y=2x$

35 .

$V.A.x=0,H.A.y=2 V.A.x=0,H.A.y=2$

37 .

$V.A.x=2,H.A.y=0 V.A.x=2,H.A.y=0$

39 .

$V.A.x=−4,H.A.y=2;( 3 2 ,0 );( 0,− 3 4 ) V.A.x=−4,H.A.y=2;( 3 2 ,0 );( 0,− 3 4 )$

41 .

$V.A.x=2,H.A.y=0,(0,1) V.A.x=2,H.A.y=0,(0,1)$

43 .

$V.A.x=−4,x= 4 3 ,H.A.y=1;(5,0);( − 1 3 ,0 );( 0, 5 16 ) V.A.x=−4,x= 4 3 ,H.A.y=1;(5,0);( − 1 3 ,0 );( 0, 5 16 )$

45 .

$V.A.x=−1,H.A.y=1;( −3,0 );( 0,3 ) V.A.x=−1,H.A.y=1;( −3,0 );( 0,3 )$; removable discontinuity (hole) at $( 1,2 )( 1,2 )$

47 .

$V.A.x=4,S.A.y=2x+9;( −1,0 );( 1 2 ,0 );( 0, 1 4 ) V.A.x=4,S.A.y=2x+9;( −1,0 );( 1 2 ,0 );( 0, 1 4 )$

49 .

$V.A.x=−2,x=4,H.A.y=1,( 1,0 );( 5,0 );( −3,0 );( 0,− 15 16 ) V.A.x=−2,x=4,H.A.y=1,( 1,0 );( 5,0 );( −3,0 );( 0,− 15 16 )$

51 .

$y=50 x 2 −x−2 x 2 −25 y=50 x 2 −x−2 x 2 −25$

53 .

$y=7 x 2 +2x−24 x 2 +9x+20 y=7 x 2 +2x−24 x 2 +9x+20$

55 .

$y= 1 2 x 2 −4x+4 x+1 y= 1 2 x 2 −4x+4 x+1$

57 .

$y=4 x−3 x 2 −x−12 y=4 x−3 x 2 −x−12$

59 .

27(x - 2) / ((x - 3)² (x + 3)) $y=27 x−2 (x−3) 2 (x+3) y=27 x−2 (x−3) 2 (x+3)$

61 .

$y= 1 3 x 2 +x−6 x−1 y= 1 3 x 2 +x−6 x−1$

63 .

$y=−6 (x−1) 2 (x+3) (x−2) 2 y=−6 (x−1) 2 (x+3) (x−2) 2$

65 .
 $x x$ 2.01 2.001 2.0001 1.99 1.999 $y y$ 100 1,000 10,000 –100 –1,000
 $x x$ 10 100 1,000 10,000 100,000 $y y$ 0.125 0.0102 .001 .0001 .00001

Vertical asymptote $x=2, x=2,$ Horizontal asymptote $y=0 y=0$

67 .
 $x x$ –4.1 –4.01 –4.001 –3.99 –3.999 $y y$ 82 802 8,002 –798 –7998
 $x x$ 10 100 1,000 10,000 100,000 $y y$ 1.4286 1.9331 1.992 1.9992 1.999992

Vertical asymptote $x=−4, x=−4,$ Horizontal asymptote $y=2 y=2$

69 .
 $x x$ –.9 –.99 –.999 –1.1 –1.01 $y y$ 81 9,801 998,001 121 10,201
 $x x$ 10 100 1,000 10,000 100,000 $y y$ 0.82645 0.9803 .998 .9998

Vertical asymptote $x=−1, x=−1,$ Horizontal asymptote $y=1 y=1$

71 .

$( 3 2 ,∞ ) ( 3 2 ,∞ )$

73 .

$(−2,1)∪(4,∞) (−2,1)∪(4,∞)$

75 .

$( 2,4 ) ( 2,4 )$

77 .

$( 2,5 ) ( 2,5 )$

79 .

$( –1,1 ) ( –1,1 )$

81 .

$C(t)= 8+2t 300+20t C(t)= 8+2t 300+20t$

83 .

85 .

$A(x)=50 x 2 + 800 x . A(x)=50 x 2 + 800 x .$ 2 by 2 by 5 feet.

87 .

$A(x)=π x 2 + 100 x . A(x)=π x 2 + 100 x .$ Radius = 2.52 meters.

### 3.8 Section Exercises

1 .

It can be too difficult or impossible to solve for $x x$ in terms of $y. y.$

3 .

We will need a restriction on the domain of the answer.

5 .

$f −1 (x)= x +4 f −1 (x)= x +4$

7 .

$f −1 (x)= x+3 −1 f −1 (x)= x+3 −1$

9 .

$f −1 (x)=− x−5 3 f −1 (x)=− x−5 3$

11 .

$f(x)= 9−x f(x)= 9−x$

13 .

$f −1 (x)= x−5 3 f −1 (x)= x−5 3$

15 .

$f −1 (x)= 4−x 3 f −1 (x)= 4−x 3$

17 .

$f −1 (x)= x 2 −1 2 ,[ 0,∞ ) f −1 (x)= x 2 −1 2 ,[ 0,∞ )$

19 .

$f −1 (x)= ( x−9 ) 2 +4 4 ,[ 9,∞ ) f −1 (x)= ( x−9 ) 2 +4 4 ,[ 9,∞ )$

21 .

$f −1 (x)= ( x−9 2 ) 3 f −1 (x)= ( x−9 2 ) 3$

23 .

$f −1 (x)= 2−8x x f −1 (x)= 2−8x x$

25 .

$f −1 (x)= 7x−3 1−x f −1 (x)= 7x−3 1−x$

27 .

$f −1 (x)= 5x−4 4x+3 f −1 (x)= 5x−4 4x+3$

29 .

$f −1 (x)= x+1 −1 f −1 (x)= x+1 −1$

31 .

$f −1 (x)= x+6 +3 f −1 (x)= x+6 +3$

33 .

$f −1 (x)= 4−x f −1 (x)= 4−x$

35 .

$f −1 (x)= x +4 f −1 (x)= x +4$

37 .

$f −1 (x)= 1−x 3 f −1 (x)= 1−x 3$

39 .

$f −1 (x)= x+8 +3 f −1 (x)= x+8 +3$

41 .

$f −1 (x)= 1 x f −1 (x)= 1 x$

43 .

$[−2,1)∪[3,∞) [−2,1)∪[3,∞)$

45 .

$[−4,2)∪[5,∞) [−4,2)∪[5,∞)$

47 .

$(–2,0);(4,2);(22,3) (–2,0);(4,2);(22,3)$

49 .

$(–4,0);(0,1);(10,2) (–4,0);(0,1);(10,2)$

51 .

$(–3,−1);(1,0);(7,1) (–3,−1);(1,0);(7,1)$

53 .

$f −1 (x)= x+ b 2 4 − b 2 f −1 (x)= x+ b 2 4 − b 2$

55 .

$f −1 (x)= x 3 −b a f −1 (x)= x 3 −b a$

57 .

$t(h)= 200−h 4.9 , t(h)= 200−h 4.9 ,$ 5.53 seconds

59 .

$r(V)= 3V 4π 3 , r(V)= 3V 4π 3 ,$ 3.63 feet

61 .

$n(C)= 100C−25 0.6−C , n(C)= 100C−25 0.6−C ,$ 250 mL

63 .

$r(V)= V 6π , r(V)= V 6π ,$ 3.99 meters

65 .

$r(V)= V 4π , r(V)= V 4π ,$ 1.99 inches

### 3.9 Section Exercises

1 .

The graph will have the appearance of a power function.

3 .

No. Multiple variables may jointly vary.

5 .

$y=5 x 2 y=5 x 2$

7 .

$y=11944 x 3 y=11944 x 3$

9 .

$y=6 x 4 y=6 x 4$

11 .

$y= 18 x 2 y= 18 x 2$

13 .

$y= 81 x 4 y= 81 x 4$

15 .

$y= 20 x 3 y= 20 x 3$

17 .

$y=10xzw y=10xzw$

19 .

$y=10x z y=10x z$

21 .

$y=4 xz w y=4 xz w$

23 .

$y=40 xz w t 2 y=40 xz w t 2$

25 .

$y=256 y=256$

27 .

$y=6 y=6$

29 .

$y=6 y=6$

31 .

$y=27 y=27$

33 .

$y=3 y=3$

35 .

$y=18 y=18$

37 .

$y=90 y=90$

39 .

$y= 81 2 y= 81 2$

41 .

$y= 3 4 x 2 y= 3 4 x 2$

43 .

$y= 1 3 x y= 1 3 x$

45 .

$y= 4 x 2 y= 4 x 2$

47 .

≈ 1.89 years

49 .

≈ 0.61 years

51 .

3 seconds

53 .

48 inches

55 .

≈ 49.75 pounds

57 .

≈ 33.33 amperes

59 .

≈ 2.88 inches

### Review Exercises

1 .

$2−2i 2−2i$

3 .

$24+3i 24+3i$

5 .

${2+i,2−i} {2+i,2−i}$

7 .

$f(x)= (x−2) 2 −9vertex(2,–9),intercepts(5,0);(–1,0);(0,–5) f(x)= (x−2) 2 −9vertex(2,–9),intercepts(5,0);(–1,0);(0,–5)$

9 .

$f(x)= 3 25 ( x+2 ) 2 +3 f(x)= 3 25 ( x+2 ) 2 +3$

11 .

300 meters by 150 meters, the longer side parallel to river.

13 .

Yes, degree = 5, leading coefficient = 4

15 .

Yes, degree = 4, leading coefficient = 1

17 .

$Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞ Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞$

19 .

–3 with multiplicity 2, $− 1 2 − 1 2$ with multiplicity 1, –1 with multiplicity 3

21 .

4 with multiplicity 1

23 .

$1 2 1 2$ with multiplicity 1, 3 with multiplicity 3

25 .

$x 2 +4 x 2 +4$with remainder 12

27 .

$x 2 −5x+20− 61 x+3 x 2 −5x+20− 61 x+3$

29 .

$2 x 2 −2x−3 2 x 2 −2x−3$, so factored form is $(x+4)(2 x 2 −2x−3) (x+4)(2 x 2 −2x−3)$

31 .

${ −2,4,− 1 2 } { −2,4,− 1 2 }$

33 .

${ 1,3,4, 1 2 } { 1,3,4, 1 2 }$

35 .

0 or 2 positive, 1 negative

37 .

Intercepts $(–2,0)and( 0,− 2 5 ) (–2,0)and( 0,− 2 5 )$, Asymptotes $x=5 x=5$ and $y=1. y=1.$

39 .

Intercepts (3, 0), (-3, 0), and $( 0, 27 2 ) ( 0, 27 2 )$, Asymptotes $x=1,x=–2,y=3. x=1,x=–2,y=3.$

41 .

$y=x−2 y=x−2$

43 .

$f −1 (x)= x +2 f −1 (x)= x +2$

45 .

$f −1 (x)= x+11 −3 f −1 (x)= x+11 −3$

47 .

$f −1 (x)= (x+3) 2 −5 4 ,x≥−3 f −1 (x)= (x+3) 2 −5 4 ,x≥−3$

49 .

$y=64 y=64$

51 .

$y=72 y=72$

53 .

148.5 pounds

### Practice Test

1 .

$20−10i 20−10i$

3 .

${2+3i,2−3i} {2+3i,2−3i}$

5 .

$Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞ Asx→−∞,f(x)→−∞,asx→∞,f(x)→∞$

7 .

$f(x)= ( x+1 ) 2 −9 f(x)= ( x+1 ) 2 −9$, vertex $( −1,−9 ) ( −1,−9 )$, intercepts $( 2,0 );( −4,0 );( 0,−8 ) ( 2,0 );( −4,0 );( 0,−8 )$

9 .

60,000 square feet

11 .

0 with multiplicity 4, 3 with multiplicity 2

13 .

$2 x 2 −4x+11− 26 x+2 2 x 2 −4x+11− 26 x+2$

15 .

$2 x 2 −x−4 2 x 2 −x−4$. So factored form is $(x+3)(2 x 2 −x−4) (x+3)(2 x 2 −x−4)$

17 .

$− 1 2 − 1 2$ (has multiplicity 2), $−1±i 15 2 −1±i 15 2$

19 .

$−2 −2$ (has multiplicity 3), $±i ±i$

21 .

$f(x)=2 ( 2x−1 ) 3 ( x+3 ) f(x)=2 ( 2x−1 ) 3 ( x+3 )$

23 .

Intercepts $(−4,0),( 0,− 4 3 ) (−4,0),( 0,− 4 3 )$, Asymptotes $x=3,x=−1,y=0 x=3,x=−1,y=0$.

25 .

$y=x+4 y=x+4$

27 .

$f −1 (x)= x+4 3 3 f −1 (x)= x+4 3 3$

29 .

$y=18 y=18$

31 .

4 seconds