Name: ______________________ Class: _________________ Date: _________
ID: A
Algebra II Chapter 6 Test (Polynomial Functions)
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
____
____
____
____
1. Classify –7x5 – 6x4 + 4x3 by degree and by number of terms.
a. quartic trinomial
c. cubic binomial
b. quintic trinomial
d. quadratic binomial
2. Zach wrote the formula w(w – 1)(5w + 4) for the volume of a rectangular prism he is designing, with
width w, which is always has a positive value greater than 1. Find the product and then classify this
polynomial by degree and by number of terms.
a. 5w 5 − w 4 − 4w 3 ; quintic trinomial
b. 20w 2 ; quadratic monomial
c. 5w 3 − w 2 − 4w; cubic trinomial
d. 5w 4 − w 3 − 4w 2 ; quartic trinomial
3. Write 4x2(–2x2 + 5x3) in standard form. Then classify it by degree and number of terms.
a. 2x + 9x4; quintic binomial
c. 2x5 – 8x4; quintic trinomial
5
4
b. 20x – 8x ; quintic binomial
d. 20x5 – 10x4; quartic binomial
4. Use a graphing calculator to determine which type of model best fits the values in the table.
x
–6
–2
0
2
6
y
–174
–26
0
–6
–114
a. linear model
c. quadratic model
b. cubic model
d. none of these
5. The table shows the number of hybrid cottonwood trees planted in tree farms in Oregon since 1995.
Find a cubic function to model the data and use it to estimate the number of cottonwoods planted in
2006.
Years since 1995
1
3
5
7
9
Trees planted (in thousands)
____
____
1.3
18.3
70.5
177.1
357.3
a. T(x) = 0.4x 3 + 0.5x 2 − 0.1x + 0.3; 630.3 thousand trees
b. T(x) = 0.4x 3 + 0.8x 2 + 0.1x; 630.3 thousand trees
c. T(x) = 0.6x 3 + 0.8x 2 − 0.1x; 618.1 thousand trees
d. T(x) = 0.6x 3 + 0.5x 2 + 0.1x + 0.3; 618.1 thousand trees
6. Write the expression (x + 6)(x – 4) as a polynomial in standard form.
a. x2 – 10x + 2
c. x2 + 2x – 24
2
b. x + 10x – 24
d. x2 + 10x – 10
7. Write 4x3 + 8x2 – 96x in factored form.
a. 6x(x + 4)(x – 4)
c. 4x(x + 6)(x + 4)
b. 4x(x – 4)(x + 6)
d. –4x(x + 6)(x + 4)
1
Name: ______________________
____
____
ID: A
8. Miguel is designing shipping boxes that are rectangular prisms. One shape of box with height h in
feet, has a volume defined by the function V(h) = h(h − 10)(h − 8). Graph the function. What is the
maximum volume for the domain 0 < h < 10? Round to the nearest cubic foot.
a. 10 ft3
b. 107 ft3
c. 105 ft3
d. 110 ft3
9. Find the zeros of y = x(x − 3)(x − 2). Then graph the equation.
a. 3, 2, –3
c. 3, 2
b.
0, –3, –2
d. 0, 3, 2
____
10. Write a polynomial function in standard form with zeros at 5, –4, and 1.
a. f(x) = x 3 − 2x 2 − 19x − 9
c. f(x) = x 3 − 21x 2 + 60x − 9
b. f(x) = x 3 − 2x 2 − 19x + 20
d. f(x) = x 3 + 20x 2 − 2x − 19
____
11. Find the zeros of f(x) = (x + 3) 2 (x − 5) 6 and state the multiplicity.
a. 2, multiplicity –3; 5, multiplicity 6
b. –3, multiplicity 2; 6, multiplicity 5
c. –3, multiplicity 2; 5, multiplicity 6
d. 2, multiplicity –3; 6, multiplicity 5
12. Divide 3x 3 − 3x 2 − 4x + 3 by x + 3.
a. 3x 2 − 12x + 32
c. 3x 2 + 6x − 40
b. 3x 2 − 12x + 32, R –93
d. 3x 2 + 6x − 40, R 99
____
2
Name: ______________________
____
ID: A
13. Determine which binomial is a factor of −2x 3 + 14x 2 − 24x + 20.
a. x + 5
b. x + 20
c. x – 24
d. x – 5
Divide using synthetic division.
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____
14. (x 4 + 15x 3 − 77x 2 + 13x − 36) ÷ (x − 4)
a. x 3 − 23x 2 − 75x − 5
b. x 3 + 15x 2 − 23x − 5
15. Use synthetic division to find P(2) for P(x)
a. 2
b. 28
c. x 3 − x 2 + 9x + 19
d. x 3 + 19x 2 − x + 9
= x 4 + 3x 3 − 6x 2 − 10x + 8.
c. 4
d. –16
Solve the equation by graphing.
____
____
16. x 2 + 7x + 19 = 0
a. x = 49
b. no solution
3
2
17. −8x − 13x + 6x = 0
a. no solution
b. –2, 0.38
c. x = 19
d. x = 12
c. 0, 2, –0.38
d. 0, –2, 0.38
Factor the expression.
____
____
____
____
____
____
18. x 3 + 216
a. (x − 6)(x 2 + 6x + 36)
c. (x − 6)(x 2 − 6x + 36)
b. (x + 6)(x 2 − 6x + 36)
d. (x + 6)(x 2 + 6x + 72)
19. c 3 − 512
a. −(c − 8)(c 2 + 8c + 64)
c. (c + 8)(c 2 + 8c + 64)
b. (c − 8)(c 2 + 8c + 64)
d. (c − 8)(c 2 − 8c − 64)
20. x 4 − 20x 2 + 64
a. (x − 2)(x − 2)(x + 4)(x + 4)
c. (x − 2)(x + 2)(x − 4)(x + 4)
2
b. (x − 2)(x − 4)(x )
d. no solution
21. Ian designed a child’s tent in the shape of a cube. The volume of the tent in cubic feet can be modeled
by the equation s 3 − 64 = 0, where s is the side length. What is the side length of the tent?
a. 4 feet
b. 16 feet
c. 64 feet
d. 8 feet
22. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation
x 3 + x 2 − 7x − 4 = 0. Do not find the actual roots.
a. –4, –2, –1, 1, 2, 4
c. 1, 2, 4
b. no roots
d. –4, –1, 1, 4
4
3
2
23. Find the rational roots of x + 8x + 7x − 40x − 60 = 0.
a. 2, 6
b. –6, –2
c. –2, 6
d. –6, 2
3
Name: ______________________
ID: A
Find the roots of the polynomial equation.
____
____
____
____
____
____
____
24. x 3
a.
b.
25. x 4
a.
b.
− 2x 2 + 10x + 136 = 0
–3 ± 5i, –4
3 ± 5i, –4
− 5x 3 + 11x 2 − 25x + 30 = 0
−2, − 3, ± i 5
2, − 3, ± 5
c. –3 ± i, 4
d. 3 ± i, 4
c. −2, 3, ± 5
d. 2, 3, ± i 5
26. A polynomial equation with rational coefficients has the roots 5 + 1 , 4 − 7 . Find two additional
roots.
a. 1 + 5 , 7 − 4
c. 5 + 1 , 4 − 7
b. 5 − 1 , 4 + 7
d. 1 − 5 , 7 + 4
27. Find a third-degree polynomial equation with rational coefficients that has roots –5 and 6 + i.
a. x 3 − 7x 2 − 23x + 185 = 0
c. x 3 − 7x 2 − 23x = 0
3
2
b. x − 7x − 12x + 37 = 0
d. x 3 − 12x 2 + 37x = 0
28. Find a quadratic equation with roots –1 + 4i and –1 – 4i.
a. x 2 − 2x + 17 = 0
c. x 2 + 2x + 17 = 0
2
b. x + 2x − 17 = 0
d. x 2 − 2x − 17 = 0
29. For the equation 2x 4 − 5x 3 + 10 = 0, find the number of complex roots and the possible number of
real roots.
a. 4 complex roots; 0, 2 or 4 real roots
b. 4 complex roots; 1 or 3 real roots
c. 3 complex roots; 1 or 3 real roots
d. 3 complex roots; 0, 2 or 4 real roots
30. Find all zeros of 2x 4 − 5x 3 + 53x 2 − 125x + 75 = 0.
a. −1, − 3 , ± 5i
c. 1, 3 , ± 5
2
2
b. 1, 3 , ± 5i
d. −1, − 3 , ± 5
2
2
4
Name: ______________________
ID: A
Short Answer
31. The table shows the population of Rockerville over a twenty-five year period. Let 0 represent 1975.
Population of Rockerville
Year
Population
1975
336
1980
350
1985
359
1990
366
1995
373
2000
395
a. Find a quadratic model for the data.
b. Find a cubic model for the data.
c. Graph each model. Compare the quadratic model and cubic model to determine which is a better fit.
32. The volume in cubic feet of a box can be expressed as V(x) = x 3 − 6x 2 + 8x, or as the product of
three linear factors with integer coefficients. The width of the box is 2 – x.
a. Factor the polynomial to find linear expressions for the height and the width.
b. Graph the function. Find the x-intercepts. What do they represent?
c. Describe a realistic domain for the function.
d. Find the maximum volume of the box.
Essay
33. A model for the height of a toy rocket shot from a platform is y = −16x 2 + 145x + 7, where x is the
time in seconds and y is the height in feet.
a. Graph the function.
b. Find the zeros of the function.
c. What do the zeros represent? Are they realistic?
d. About how high does the rocket fly before hitting the ground? Explain.
34. Find the rational roots of 4x 3 − 3x − 1 = 0. Explain the process you use and show your work.
5
ID: A
Algebra II Chapter 6 Test (Polynomial Functions)
Answer Section
MULTIPLE CHOICE
1. ANS:
REF:
STA:
KEY:
2. ANS:
REF:
STA:
KEY:
3. ANS:
REF:
STA:
KEY:
4. ANS:
REF:
OBJ:
STA:
KEY:
5. ANS:
REF:
OBJ:
STA:
KEY:
6. ANS:
REF:
OBJ:
STA:
KEY:
7. ANS:
REF:
OBJ:
STA:
KEY:
8. ANS:
REF:
OBJ:
STA:
KEY:
9. ANS:
REF:
OBJ:
STA:
KEY:
B
PTS: 1
DIF: L2
6-1 Polynomial Functions
OBJ: 6-1.1 Exploring Polynomial Functions
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 1
degree of a polynomial | polynomial
C
PTS: 1
DIF: L3
6-1 Polynomial Functions
OBJ: 6-1.1 Exploring Polynomial Functions
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 1
degree of a polynomial | polynomial
B
PTS: 1
DIF: L3
6-1 Polynomial Functions
OBJ: 6-1.1 Exploring Polynomial Functions
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 1
degree of a polynomial | polynomial | standard form of a polynomial
C
PTS: 1
DIF: L2
6-1 Polynomial Functions
6-1.2 Modeling Data with a Polynomial Function
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 2
polynomial function | modeling data | graphing calculator
B
PTS: 1
DIF: L2
6-1 Polynomial Functions
6-1.2 Modeling Data with a Polynomial Function
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 3
modeling data | polynomial function | cubic function | graphing calculator
C
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.1 The Factored Form of a Polynomial
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 1
polynomial | standard form of a polynomial
B
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.1 The Factored Form of a Polynomial
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 2
factoring a polynomial | polynomial
C
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.1 The Factored Form of a Polynomial
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 3
factoring a polynomial | modeling data | polynomial function | x-intercept
D
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.2 Factors and Zeros of a Polynomial Function
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 4
Zero Product Property | polynomial function | zeros of a polynomial function | graphing
1
ID: A
10. ANS:
REF:
OBJ:
STA:
KEY:
11. ANS:
REF:
OBJ:
STA:
KEY:
12. ANS:
REF:
TOP:
13. ANS:
REF:
TOP:
KEY:
14. ANS:
REF:
TOP:
KEY:
15. ANS:
REF:
TOP:
KEY:
16. ANS:
REF:
STA:
KEY:
17. ANS:
REF:
STA:
KEY:
18. ANS:
REF:
STA:
KEY:
19. ANS:
REF:
STA:
KEY:
20. ANS:
REF:
STA:
KEY:
B
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.2 Factors and Zeros of a Polynomial Function
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 5
polynomial function | standard form of a polynomial | zeros of a polynomial function
C
PTS: 1
DIF: L2
6-2 Polynomials and Linear Factors
6-2.2 Factors and Zeros of a Polynomial Function
CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 6
polynomial function | zeros of a polynomial function | multiplicity | multiple zero
B
PTS: 1
DIF: L2
6-3 Dividing Polynomials
OBJ: 6-3.1 Using Long Division
6-3 Example 1
KEY: polynomial | division of polynomials
D
PTS: 1
DIF: L2
6-3 Dividing Polynomials
OBJ: 6-3.1 Using Long Division
6-3 Example 2
division of polynomials | factoring a polynomial | polynomial
D
PTS: 1
DIF: L3
6-3 Dividing Polynomials
OBJ: 6-3.2 Using Synthetic Division
6-3 Example 3
division of polynomials | polynomial | synthetic division
C
PTS: 1
DIF: L2
6-3 Dividing Polynomials
OBJ: 6-3.2 Using Synthetic Division
6-3 Example 5
division of polynomials | polynomial | synthetic division
B
PTS: 1
DIF: L4
6-4 Solving Polynomial Equations OBJ: 6-4.1 Solving Equations by Graphing
CO 2.5 | CO 2.6
TOP: 6-4 Example 1
graphing | graphing calculator | solving equations | no solutions | polynomial function
D
PTS: 1
DIF: L2
6-4 Solving Polynomial Equations OBJ: 6-4.1 Solving Equations by Graphing
CO 2.5 | CO 2.6
TOP: 6-4 Example 1
graphing | graphing calculator | solving equations | polynomial
B
PTS: 1
DIF: L2
6-4 Solving Polynomial Equations OBJ: 6-4.2 Solving Equations by Factoring
CO 2.5 | CO 2.6
TOP: 6-4 Example 3
polynomial | factoring a polynomial
B
PTS: 1
DIF: L2
6-4 Solving Polynomial Equations OBJ: 6-4.2 Solving Equations by Factoring
CO 2.5 | CO 2.6
TOP: 6-4 Example 3
factoring a polynomial | polynomial
C
PTS: 1
DIF: L2
6-4 Solving Polynomial Equations OBJ: 6-4.2 Solving Equations by Factoring
CO 2.5 | CO 2.6
TOP: 6-4 Example 5
factoring a polynomial | polynomial
2
ID: A
21. ANS: A
PTS: 1
DIF: L3
REF: 6-4 Solving Polynomial Equations OBJ: 6-4.2 Solving Equations by Factoring
STA: CO 2.5 | CO 2.6
TOP: 6-4 Example 3
KEY: factoring a polynomial | polynomial function | problem solving
22. ANS: A
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 1
KEY: polynomial function | root of a function | solving equations | Rational Root Theorem
23. ANS: B
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 1
KEY: polynomial function | Rational Root Theorem | root of a function | solving equations
24. ANS: B
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 2
KEY: polynomial function | Rational Root Theorem | solving equations | root of a function
25. ANS: D
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 2
KEY: polynomial function | Rational Root Theorem | solving equations | root of a function
26. ANS: B
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.2 Irrational Root Theorem and Imaginary Root Theorem
TOP: 6-5 Example 3
KEY: polynomial function | solving equations | Irrational Root Theorem | conjugates
27. ANS: A
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.2 Irrational Root Theorem and Imaginary Root Theorem
TOP: 6-5 Example 5
KEY: Imaginary Root Theorem | conjugates | polynomial function | root of a function
28. ANS: C
PTS: 1
DIF: L2
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.2 Irrational Root Theorem and Imaginary Root Theorem
TOP: 6-5 Example 5
KEY: Imaginary Root Theorem | conjugates | polynomial function | root of a function
29. ANS: A
PTS: 1
DIF: L2
REF: 6-6 The Fundamental Theorem of Algebra
OBJ: 6-6.1 The Fundamental Theorem of Algebra
TOP: 6-6 Example 1
KEY: Fundamental Theorem of Algebra | Imaginary Root Theorem | Rational Root Theorem | root
of a function | polynomial function
30. ANS: B
PTS: 1
DIF: L2
REF: 6-6 The Fundamental Theorem of Algebra
OBJ: 6-6.1 The Fundamental Theorem of Algebra
TOP: 6-6 Example 2
KEY: Fundamental Theorem of Algebra | Rational Root Theorem | polynomial function | root of a
function | zeros of a polynomial function
3
ID: A
SHORT ANSWER
31. ANS:
a. y = 0.023x 2 + 1.549x + 338.571
b. y = 0.0079x 3 − 0.2716x 2 + 4.2378x + 335.6270
c.
The cubic model is a better fit.
PTS:
OBJ:
STA:
KEY:
32. ANS:
a. V(x)
b.
1
DIF: L3
REF: 6-1 Polynomial Functions
6-1.2 Modeling Data with a Polynomial Function
CO 2.7 | CO 2.1 | CO 2.2
TOP: 6-1 Example 3
cubic function | quadratic function | graphing calculator | modeling data | multi-part question
= x(2 − x)(4 − x)
x-intercepts: x = 0, 2, 4. These are the values of x that produce a volume of 0.
c. 0 < x < 2
d. 3.08 cubic feet
PTS: 1
DIF: L2
REF: 6-2 Polynomials and Linear Factors
OBJ: 6-2.1 The Factored Form of a Polynomial
STA: CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 3
KEY: factoring a polynomial | graphing calculator | polynomial function | x-intercept | problem
solving | multi-part question | word problem
4
ID: A
ESSAY
33. ANS:
[4] a.
b.
c.
x ≈ –0.05, x ≈ 9.11
The zeros represent the times at which the height of the rocket is 0. The time
–0.05 seconds is not realistic. The time 9.11 seconds is the time at which
the rocket lands.
d.
about 336 feet; The height is the maximum value of the function.
[3] an error in one of the three parts of the question
[2] an error in two parts of the question
[1] one part missing and errors in answer or reasoning for one of the other parts
PTS: 1
DIF: L3
REF: 6-2 Polynomials and Linear Factors
OBJ: 6-2.1 The Factored Form of a Polynomial
STA: CO 2.3 | CO 2.1 | CO 2.2 | CO 2.7 TOP: 6-2 Example 3
KEY: reasoning | graphing | graphing calculator | modeling data | polynomial function | problem
solving | relative maximum | x-intercept | zeros of a polynomial function | extended response |
rubric-based question | writing in math | word problem
5
ID: A
34. ANS:
[4] Step 1:
List the possible rational roots by using the Rational Root Theorem. The leading
coefficient is 4 with factors of ±1, ±2, and ±4. The constant term is –1 with
factors of –1 and 1. The only possible roots of the equation have the form
factor of − 1 . Those roots would be ±1, ± 1 , and ± 1 .
factor of 4
2
4
Step 2:
Test each possible rational root in the equation. The only roots that satisfy the
equation are − 1 and 1.
2
[3] an error in computation or missing part of the explanation
[2] several errors in computation or in the explanation
[1] one root given with no explanation
PTS: 1
DIF: L4
REF: 6-5 Theorems About Roots of Polynomial Equations
OBJ: 6-5.1 The Rational Root Theorem TOP: 6-5 Example 1
KEY: extended response | polynomial function | Rational Root Theorem | root of a function |
rubric-based question | writing in math
6