Section 5.1 Polynomial functions
#1 – 22
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
c) Determine the maximum number of turning points on the graph
d) Which function will the graph of f(x) behave like for large values of |𝑥|
e) Describe the end behavior
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
1) 𝑓(𝑥) = (𝑥 − 3)2 (𝑥 + 1)
2) 𝑓(𝑥) = (𝑥 − 5)2 (𝑥 + 2)
3) 𝑓(𝑥) = (2𝑥 − 3)2 (𝑥 − 1)
4) 𝑓(𝑥) = (3𝑥 − 2)2 (𝑥 + 1)5
5) 𝑓(𝑥) = (𝑥 2 − 3)2 (2𝑥 + 1)
6) 𝑓(𝑥) = (𝑥 2 − 5)2 (2𝑥 + 5)
7) 𝑓(𝑥) = (𝑥 2 − 6)2 (2𝑥 − 12)2
8) 𝑓(𝑥) = (𝑥 2 − 4)(2𝑥 − 10)2
9) 𝑓(𝑥) = (𝑥 2 + 3)(2𝑥 + 1)2
10) 𝑓(𝑥) = (𝑥 2 + 5)(2𝑥 + 5)
11) 𝑓(𝑥) = (𝑥 2 + 6)(2𝑥 − 12)2
12) 𝑓(𝑥) = (𝑥 2 + 2)(2𝑥 + 1)2
13) 𝑓(𝑥) = 𝑥 2 + 6𝑥 − 7
14) 𝑓(𝑥) = 𝑥 2 − 6𝑥 − 7
15) 𝑓(𝑥) = 𝑥 2 − 4
16) 𝑓(𝑥) = 𝑥 2 − 9
17) 𝑓(𝑥) = −5𝑥 3 − 13𝑥 2 − 6𝑥
18) 𝑓(𝑥) = 6𝑥 3 − 4𝑥 2 − 10𝑥
19) 𝑓(𝑥) = −3𝑥 4 + 12𝑥 2
20) 𝑓(𝑥) = 2𝑥 4 − 18𝑥 2
21) 𝑓(𝑥) = 5𝑥 4 + 13𝑥 3 + 6𝑥 2
22) 𝑓(𝑥) = 6𝑥 4 − 4𝑥 3 − 10𝑥 2
#23 – 32: Form a polynomial whose x-intercepts (zeros) are given. Answers can vary depending on
choice of a leading coefficient. (multiply out your polynomial)
23) x-intercepts: 2, 3, -4; degree 3
24) x-intercepts -2, 4, 6; degree 3
1
25) x-intercepts: 2, 3, 2; degree 3
27) x-intercepts:
−1
,
3
3 ; degree 2
2
26) x-intercepts -2, 4, 3; degree 3
28) x-intercepts
3
4
,
−1
;
2
degree 2
29) x-intercepts ±√2; degree 2
30) x-intercepts ±√3; degree 2
31) x-intercepts ±√2 , 5; degree 3
32) x-intercepts ±√3 , −4; degree 3
Section 5.2: synthetic division
#1- 10:
a) Perform the division using synthetic division.
b) if the remainder is 0 use the result to completely factor the dividend (this is the numerator or the
polynomial to the left of the division sign.)
1)
3𝑥 3 −17𝑥 2 +15𝑥−25
𝑥−5
2)
5𝑥 3 +18𝑥 2 +7𝑥−6
𝑥+3
3)
4𝑥 3 +8𝑥 2 −9𝑥−18
𝑥+2
4)
9𝑥 3 −18𝑥 2 −16𝑥+32
.
𝑥−2
5)
3𝑥 3 −16𝑥 2 −72
𝑥−6
6)
5𝑥 3 −6𝑥 2 +8
𝑥−4
7)
(5𝑥 3 + 6𝑥 + 8) ÷ (𝑥 + 2)
8) (𝑥 3 + 512) ÷ (𝑥 + 8)
9) (𝑥 3 − 27) ÷ (𝑥 − 3)
10) (𝑥 3 + 5𝑥 2 ) ÷ (𝑥 + 5)
#11 – 20:
a) use your graphing calculator, or the rational root theorem to find a zero of the polynomial
i) you need to find one zero for a third degree polynomial
ii) you need to find two zeros for a fourth degree polynomial
b) use synthetic division to completely factor the polynomial (use “double” synthetic division for fourth
degree polynomials)
11) f(x) = x3 + 2x2 – 5x – 6
12) f(x) = x3 + 8x2 + 11x -20
13) f(x) = 2x3 – 13x2 + 24x – 9
14) f(x) = 2x3 – 5x2 – 4x + 12
15) f(x) = x4 + x3 – 3x2 – x + 2
16) f(x) = x4 – x3 – 6x2 + 4x+ 8
17) f(x) = 2x4 + 17x3 + 35x2 – 9x – 45
18) f(x) = 4x4 – 15x3 - 8x2 + 15x + 4
19) f(x) = x4 + 7x2 – 8
20) f(x) = x4 - 25x2 +144
#21-30: Solve
21) x4 – x3 + 2x2 – 4x – 8 = 0
22) 2x3 + 3x2 - 2x - 3 = 0
23) 4x3 +8x2 -9x-18 = 0
24) 2x3 – 3x2 – 3x +2 = 0
25) 2x4 +7x3 -4x2 -27x-18 = 0
26) 2x3 – 11x2 + 10x + 8 = 0
27) x4 +4x3 + 2x2 – x + 6 = 0
28) x4 – 2x3 +10x2 – 18x + 9 = 0
29) x4 + 7x2 – 8 = 0
30) x4 + 8x2 – 9 = 0
Section 5.3:
#1 – 10: Create a function with lead coefficient 1 that satisfies the conditions.
1)
3)
5)
7)
9)
degree 3; zeros -4 and 6 + i
degree 3; zeros 2 and 3 - i
degree 4; zeros 2i, and 5 – i
degree 4; zeros 3i, and 5i
degree 4; zeros 2i, 3, - 4
2) degree 3: zeros – 5 and 3 + i
4) degree 3: zeros 7 and 2 – i
6) degree 4; zeros 3i and 4 – i
8) degree 4; zeros 3i and 7i
10) degree 4; zeros 3i, 5 and -6
#11- 24: Use the given solutions to find the remaining solutions to the problem.
11) x3 – 4x2 + 4x – 16 = 0 (x = 2i is a solution)
12) x3 + 3x2 + 25x + 75 (x = 5i is a solution)
13) 2x4 + 5x3 +5x2 +20x-12 = 0 (2i is a solution)
14) OMIT
15) x3 -7x2 – x + 87 = 0 (5 + 2i is a solution)
16) x4 – 7x3 + 14x2 -38x-60= 0 (1+3i is a solution)
17) x4 +6x3 + 2x2 – 26x + 17 = 0 (-4+I is a solution)
18) x4 -14x3 +77x2 -200x+208 = 0 (3 + 2i is a solution)
19) x3 -8x2 +25x-26 = 0 (3+2i is a solution)
20) x3 +13x2 +57x+85=0 (-4 + I is a solution)
Section 5.4: Properties of rational functions
#1-16:
a) find the domain, express your answer using words
b) find the equation of the vertical asymptote (if any)
1) 𝑓(𝑥) =
2𝑥−6
𝑥+3
2) 𝑓(𝑥) =
3𝑥−12
2𝑥+6
3) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
5) 𝑓(𝑥) =
3𝑥 2 −10𝑥−8
𝑥 2 −4
6) 𝑓(𝑥) =
3𝑥 2 +20𝑥+12
𝑥 2 −16
7) 𝑓(𝑥) =
9) 𝑓(𝑥) =
3
𝑥−6
10) 𝑓(𝑥) = 𝑥−8
13) 𝑓(𝑥) =
𝑥+2
𝑥 2 +16
3𝑥+12
2
14) 𝑓(𝑥) =
2𝑥
3𝑥−12
𝑥 2 +25
2𝑥−6
𝑥+3
3𝑥+12
19) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
18) 𝑓(𝑥) =
5𝑥
𝑥 2 +9
16) 𝑓(𝑥) =
15) 𝑓(𝑥) =
3𝑥−12
2𝑥+6
3𝑥−18
20) 𝑓(𝑥) = 𝑥 2 +5𝑥−6
3𝑥 2 −10𝑥−8
𝑥 2 −4
22) 𝑓(𝑥) =
23) 𝑓(𝑥) =
𝑥 2 −16
𝑥 3 −𝑥 2
24) 𝑓(𝑥) = 𝑥 3 +5𝑥2 −6𝑥
25) 𝑓(𝑥) =
3
𝑥−6
26) 𝑓(𝑥) =
2
𝑥−8
28) 𝑓(𝑥) =
3𝑥
𝑥 2 −25
30) 𝑓(𝑥) =
3𝑥−12
𝑥 2 +25
2𝑥
𝑥+2
29) 𝑓(𝑥) = 𝑥 2 +16
5𝑥
31) 𝑓(𝑥) = 𝑥 2 +9
3𝑥 2 +20𝑥+12
𝑥 2 −16
3𝑥−18
2𝑥
3𝑥−18
𝑥 3 +5𝑥 2 −6𝑥
12) 𝑓(𝑥) =
21) 𝑓(𝑥) =
27) 𝑓(𝑥) = 𝑥 2 −9
8) 𝑓(𝑥) =
11) 𝑓(𝑥) = 𝑥 2 −9
#1 7– 32
a) find the x-intercept
b) find the y-intercept
17) 𝑓(𝑥) =
𝑥 2 −16
𝑥 3 −𝑥 2
3𝑥−18
4) 𝑓(𝑥) = 𝑥 2 +5𝑥−6
32) 𝑓(𝑥) = 𝑥 2 +1
3𝑥
𝑥 2 −25
2𝑥
𝑥 2 +1
Section 5.4: Properties of rational functions
#33-48: find the equation of the horizontal asymptote
33) 𝑓(𝑥) =
2𝑥−6
𝑥+3
3𝑥+12
35) 𝑓(𝑥) = 𝑥 2 −3𝑥−4
34) 𝑓(𝑥) =
3𝑥−12
2𝑥+6
3𝑥−18
36) 𝑓(𝑥) = 𝑥 2 +5𝑥−6
37) 𝑓(𝑥) =
3𝑥 2 −10𝑥−8
𝑥 2 −4
38) 𝑓(𝑥) =
39) 𝑓(𝑥) =
𝑥 2 −16
𝑥 2 −𝑥
40) 𝑓(𝑥) = 𝑥 3 +5𝑥2 −6𝑥
41) 𝑓(𝑥) =
3
𝑥−6
42) 𝑓(𝑥) =
2
𝑥−8
44) 𝑓(𝑥) =
3𝑥
𝑥 2 −25
46) 𝑓(𝑥) =
3𝑥−12
𝑥 2 +25
2𝑥
43) 𝑓(𝑥) = 𝑥 2 −9
𝑥+2
45) 𝑓(𝑥) = 𝑥 2 +16
5𝑥 2
47) 𝑓(𝑥) = 𝑥 2 +9
3𝑥 2 +20𝑥+12
𝑥 2 −16
3𝑥−18
2𝑥
48) 𝑓(𝑥) = 𝑥 2 +1
#49 – 58: find the equation of the slant asymptote
49) 𝑓(𝑥) =
𝑥 2 +5𝑥+1
2𝑥+3
50) 𝑓(𝑥) =
8𝑥 2 +3𝑥+2
4𝑥−5
51) 𝑓(𝑥) =
12𝑥 2 +5𝑥+1
4𝑥−5
52) 𝑓(𝑥) =
8𝑥 2 −6𝑥−3
2𝑥+5
53) 𝑓(𝑥) =
2𝑥 2
4𝑥−1
54) 𝑓(𝑥) =
3𝑥 2
12𝑥−5
55) 𝑓(𝑥) =
2𝑥 3 +3𝑥 2 −5
6𝑥 2 +6𝑥−1
56) 𝑓(𝑥) =
8𝑥 3 +4𝑥 2 −3𝑥−1
6𝑥 2 +5𝑥−4
𝑥3
57) 𝑓(𝑥) = 𝑥 2 +1
𝑥3
58) 𝑓(𝑥) = 𝑥 2 +3𝑥−4
Section 5.5: the graph of a rational function
For each problem find the following:
a) Domain
b) Vertical Asymptote (if any)
c) Horizontal asymptote, or slant asymptote
d) x- intercept(s) if any
e) y-intercept(s) if any
f) Sketch a graph of the function
1. f ( x) 
3x  6
x6
4x 2  9
3. f ( x)  2
x 1
5. f ( x ) 
7.
3
5x
f ( x) 
9. f ( x) 
4 x  12
2x  2
x2  4
4. f ( x) 
4x 2  1
4
6. 𝑓(𝑥) = 3𝑥
x3
x 2  16
3x 2  8 x  5
x5
11) 𝑓(𝑥) =
2. f ( x) 
2𝑥 2
4𝑥−1
8. f ( x) 
x 1
x  4 x  12
10. f ( x) 
2
2 x 2  5x  3
x3
3𝑥 2
12) 𝑓(𝑥) = 12𝑥−5
13) f ( x) 
3 x  12
2x  6
14) f ( x) 
3
x5
15) f ( x ) 
1
x
16) f ( x) 
x2
x3
17) f ( x ) 
x5
x2
18) 𝑓(𝑥) =
3𝑥
𝑥 2 −25
Section 5.6: Quadratic and rational inequalities
Solve
1) x2 + 6x – 7 < 0
2) x2 + 5x – 6 < 0
3) x2 – 6x + 5 < 0
4) x2 – 4x + 3 < 0
5) 𝑥 2 + 9𝑥 − 10 ≤ 0
6) 𝑥 2 + 2𝑥 − 8 ≤ 0
7) 𝑥 2 − 5𝑥 − 6 ≤ 0
8) 𝑥 2 − 4𝑥 − 5 ≤ 0
9) x2 + 6x – 7 > 0
10) x2 + 5x – 6 > 0
11) x2 – 6x + 5 > 0
12) x2 – 4x + 3 > 0
13) 𝑥 2 + 9𝑥 − 10 ≥ 0
14) 𝑥 2 + 2𝑥 − 8 ≥ 0
15) 𝑥 2 − 5𝑥 − 6 ≥ 0
16) 𝑥 2 − 4𝑥 − 5 ≥ 0
17)
𝑥−3
𝑥−6
<0
18)
𝑥−2
𝑥−9
<0
19)
𝑥+5
𝑥−6
<0
20)
𝑥+2
𝑥−4
<0
21)
𝑥−3
𝑥−7
>0
22)
𝑥−1
𝑥−5
>0
23)
𝑥+1
𝑥+5
>0
24)
𝑥+2
𝑥+6
>0
25)
𝑥−4
𝑥+2
<0
26)
𝑥−5
𝑥+1
<0
Chapter 5 review
For problems 1 – 3:
a) List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
b) Determine whether the graph crosses or touches the x-axis at each x-intercept
c) Determine the maximum number of turning points on the graph
d) Which function will the graph of f(x) behave like for large values of |𝑥|
e) Describe the end behavior
f) sketch a graph and approximate the turning points, also label the x-intercepts
g) state the intervals where the function is increasing and decreasing
1) 𝑓(𝑥) = (𝑥 − 5)2 (𝑥 − 3)
2) 𝑓(𝑥) = (𝑥 2 − 16)(3𝑥 − 21)
3) 𝑓(𝑥) = 𝑥 4 − 4𝑥 3 − 21𝑥 2
For problems 4 – 5:
a) use your graphing calculator, or the rational root theorem to find a zero of the polynomial
i) you need to find one zero for a third degree polynomial
ii) you need to find two zeros for a fourth degree polynomial
b) use synthetic division to completely factor the polynomial (use “double” synthetic division for fourth
degree polynomials)
4) f(x) = x3 + x2 – 9x– 9
5) f(x) = x4 +3x3 -8x2 -12x +16
#6-7: Solve
6) x4 – 13x2 + 36 = 0
7) x3 + x2 + 2x + 2= 0
#8-9: Create a function with lead coefficient 1 that satisfies the conditions.
8) degree 3; zeros -2 and 3 + i
9) degree 3: zeros – 5 and 4i
10 – 12 For each problem find the following:
a) Domain
b) Vertical Asymptote (if any)
c) Horizontal asymptote, or slant asymptote
d) x- intercept(s) if any
e) y-intercept(s) if any
f) Sketch a graph of the function : label all the features found in parts b - e
10. f ( x ) 
2x  6
x3
11. f ( x) 
x 1
2
x  4 x  21
12. f ( x) 
2x 2  7x  5
x5
#13 – 16: Solve
13) x2 + 6x – 16 < 0
16)
𝑥+4
𝑥−2
>0
14) x2 + 5x – 24 > 0
15)
𝑥−4
𝑥−6
<0
Grima MAT 151
1)
a)
b)
c)
d)
e)
Chapter 5 practice test
f(x)=x3 + 2x2 – 8x
List each x-intercept (zero) and its multiplicity (round to 2 decimal places when needed)
Which function will the graph of f(x) behave like for large values of |𝑥|
Describe the end behavior
sketch a graph and approximate the turning points, also label the x-intercepts
state the intervals where the function is increasing and decreasing (round to 2 decimals)
2) f(x) = 2x3 - 3x2 – 17x - 12
a) use your graphing calculator, or the rational root theorem to find a zero of the polynomial
b) use synthetic division to completely factor the polynomial
3) Solve: 2x3 - 3x2 – 17x - 12= 0 (hint use your answer to question 2b)
4) Create a function with lead coefficient 1 that satisfies the conditions;
degree 3: zeros 6 and 5i
5) Solve: x2 + 3x – 10 < 0
6) f ( x ) 
a)
b)
c)
d)
e)
f)
Domain
Vertical Asymptote (if any)
Horizontal asymptote, or slant asymptote
x- intercept(s) if any
y-intercept(s) if any
Sketch a graph of the function : label all the features found in parts b - e
7. f ( x) 
a)
b)
c)
d)
e)
f)
2x  8
x2
x  10
x  4x  5
2
Domain
Vertical Asymptote (if any)
Horizontal asymptote, or slant asymptote
x- intercept(s) if any
y-intercept(s) if any
Sketch a graph of the function : label all the features found in parts b – e
1a) (-4,0) odd (0,0) odd (2,0) odd
1b) like x3
1c) falls to the left, rises to the right
1d)
1e) increasing (−∞, −2.43) ∪ (1.09, ∞) decreasing (-2.43, 1.09)
2a) x = 4 (also x = -1)
2b) f(x) = (x-4)(x+1)(2x+3)
3) x = 4, -1, -3/2
4) f(x) = x3 – 6x2 + 25x - 150
5) -5 < x < 2
6a)
6b)
6c)
6d)
6e)
Domain all real numbers except 2
x =2
y=2
(-4,0)
(0, -4)
6f)
7a)
7b)
7c)
7d)
7e)
7f)
Domain all real numbers except x = -5,1
x = -5 and x = 1
y=0
(-10, 0)
(0,-2)