Mastering Polynomials
Name ___________________________________________
Find all of the real and imaginary zeros for the polynomial function.
1) f(x) = x3 - 4x2 - 36x + 144
2) f(x) = x4 + 15x3 + 49x2 - 15x - 50
Describe the behavior of the function's graph at its x-intercepts.
3) f(x) = (x - 3)2(x + 8)2
A) Crosses at (3, 0) but not at (-8, 0)
C) Crosses at (8, 0) but not at (-3, 0)
4) x3 + 2x2 - 4x - 8
A) Does not cross at (-2, 0) or (2, 0)
C) Crosses at (-2, 0) and (2, 0)
B) Crosses at (3, 0) and (-8, 0)
D) Does not cross at (3, 0) or (-8, 0)
B) Crosses at (-2, 0), does not cross at (2, 0)
D) Does not cross at (-2, 0), crosses at (2, 0)
Solve the problem.
5) The polynomial G(x) = -0.006x4 + 0.140x 3 - 0.53x2 + 1.79x measures the concentration of a dye in the
bloodstream x seconds after it is injected. Does the concentration increase between 10 and 11 seconds?
A) No
B) Yes
6) A piece of rectangular sheet metal is 22 inches wide. It is to be made into a rain gutter by turning up equal
edges to form parallel sides. Let x represent the length of each of the parallel sides. For what value of x will
the area of the cross section be a maximum (and thus maximize the amount of water that the gutter will
hold)? If necessary, round to 2 decimal places.
A) 44
B) 5.5
C) 11
D) 88
7) The polynomial 0.0041x3 + 0.0046x2 + 0.143x + 1.08 gives the approximate total earnings of a company, in
millions of dollars, where x represents the number of years since 1996. This model is valid for the years from
1996 to 2000. Determine the earnings for 1997.
A) $1.22 million
B) $1.08 million
C) $1.23 million
D) $1.42 million
Sketch the graph of the polynomial function.
8) P(x) = 2x(x + 1)2
y
6x
Determine whether the given number is a zero of the polynomial function.
9) P(x) = -4x3 + x2 + 7x - 8; 3
Sketch the graph of the polynomial function.
10) f(x) = 2x(x + 1)(x - 2)
y
6
6x
-6
Use the rational zero theorem to find all possible rational zeros for the polynomial function.
11) P(x) = 2x3 + 6x2 + 13x - 8
12) P(x) = -2x4 + 5x3 + 4x2 + 18
Use the factor theorem to decide whether or not the second polynomial is a factor of the first.
13) 5x4 + 16x3 - 3x2 + x + 4; x + 3
14) 6x4 + 11x3 - 2x2 + x + 2; x + 2
Describe the correct end behavior for the given polynomial function.
15) P(x) = 1.48x4 + 3x 2 + x - 8
16) P(x) = -x5 - 5x3 - 3x + 3
Find the x- and y-intercepts of f.
17) f(x) = 5x - x3
18) f(x) = -x2(x + 9)(x2 + 1)
Find the zeros of the polynomial function and state the multiplicity of each.
19) f(x) = x 3 + 4x2 - x - 4
20) f(x) = (x + 2)2(x - 1)
Find the real solutions of the equation.
21) 3x3 - 14x2 + 13x + 6 = 0
Form a polynomial whose zeros and degree are given.
22) Zeros: 3, multiplicity 2; -3, multiplicity 2; degree 4
23) The graph of f(x) = -x4 + x3 + 8x2 - 12x is shown below. Use the graph to factor f(x).
y
25
-5 -4 -3 -2 -1
1
2
3
4
5 x
-25
Solve the problem.
24) The graph of f(x) = x4 + x3 - 6x2 - x2 - x + 6 is shown below. Use the graph to factor f(x).
y
15
-5 -4 -3 -2 -1
1
2
3
4
5 x
-15
A) f(x) = -(x + 3)(x + 1)(x - 1)(x - 2)
C) f(x) = (x - 3)(x - 1)(x + 1)(x + 2)
B) f(x) = (x + 3)(x + 1)(x - 1)(x - 2)
D) f(x) = x(x + 3)(x + 1)(x - 1)(x - 2)
Find the equation and sketch the graph of the function.
25) A quadratic function with x-intercepts (-1, 0) and (- 5, 0) and y-intercept (0, 5).
y
10
5
-10
-5
5
-5
-10
10 x
Answer Key
Testname: UNIT 5 ASSESSMENT_NOCALCA
1) -6, 4, 6
2) -10, -5, -1, 1
3) D
4) D
5) B
6) B
7) C
8)
y
6x
9) No
10)
y
6x
11) ±1, ± 1 , ±2, ±4, ±8
2
12) ±1, ± 1 , ±2, ±3, ± 3 , ±6, ±9, ± 9 , ±18
2
2
2
13) No
14) Yes
15)
16)
17) x-intercepts: 0, 5, - 5; y-intercept: 0
18) x-intercepts: -9, 0; y-intercept: 0
19) -4, multiplicity 1; -1, multiplicity 1; 1, multiplicity 1
20) -2, multiplicity 2; 1, multiplicity 1
1
21) - , 2, 3
3
22) f(x) = x4 - 18x2 + 81
Answer Key
Testname: UNIT 5 ASSESSMENT_NOCALCA
23) f(x) = -x(x + 3)(x - 2)2
24) B
25) y = x2 + 6x + 5
y
10
5
-10
-5
5
-5
-10
10 x