Download Polynomial Review Answer Section

Polynomial Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Use a graphing calculator to determine which type of model best fits the values in the table.
x
–6
–2
0
2
6
y
–6
–2
0
2
6
a. quadratic model
b. cubic model
____
____
____
c. linear model
d. none of these
2. Determine which binomial is not a factor of
a. x + 4
b. x + 3
c. x – 5
d. 4x + 3
.
3. Determine which binomial is a factor of
a. x + 5
b. x + 20
c. x – 24
.
d. x – 5
4. The volume of a shipping box in cubic feet can be expressed as the polynomial
. Each
dimension of the box can be expressed as a linear expression with integer coefficients. Which expression
could represent one of the three dimensions of the box?
a. x + 6
c. 2x + 3
b. x + 1
d. 2x + 1
Short Answer
5. 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.
6. The volume in cubic feet of a box can be expressed as
factors with integer coefficients. The width of the box is 2 – x.
, or as the product of three linear
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.
7. The volume in cubic feet of a workshop’s storage chest can be expressed as the product of its three
dimensions:
. The depth is x + 1.
a. Find linear expressions with integer coefficients for the other dimensions.
b. If the depth of the chest is 6 feet, what are the other dimensions?
8. Classify –3x5 – 2x3 by degree and by number of terms.
9. Classify –7x5 – 6x4 + 4x3 by degree and by number of terms.
10. 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.
11. Determine the probability of getting four heads when tossing a coin four times.
12. Use the Binomial Theorem to expand
.
13. A manufacturer of shipping boxes has a box shaped like a cube. The side length is
5a + 4b. What is the volume of the box in terms of a and b?
Use Pascal’s Triangle to expand the binomial.
14.
15.
16. Find all zeros of
.
For the equation, find the number of complex roots, the possible number of real roots, and the possible
rational roots.
17.
18.
19. For the equation
roots.
, find the number of complex roots and the possible number of real
20. Find a quadratic equation with roots –1 + 4i and –1 – 4i.
21. Find a third-degree polynomial equation with rational coefficients that has roots –5 and 6 + i.
22. A polynomial equation with rational coefficients has the roots
. Find two additional roots.
Find the roots of the polynomial equation.
23.
24.
25. Use the Rational Root Theorem to list all possible rational roots of the polynomial equation
. Do not find the actual roots.
26. Solve
.
27. 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
, where s is the side length. What is the side length of the tent?
28. Solve
. Find all complex roots.
Factor the expression.
29.
30.
31. Over two summers, Ray saved $1000 and $600. The polynomial
represents her savings after
three years, where x is the growth factor. (The interest rate r is x – 1.) What is the interest rate she needs to
save $1850 after three years?
32. The dimensions in inches of a shipping box at We Ship 4 You can be expressed as width x, length x + 5, and
height 3x – 1. The volume is about 7.6 ft3. Find the dimensions of the box in inches. Round to the nearest
inch.
Solve the equation by graphing.
33.
34.
35. Use synthetic division to find P(2) for
.
Divide using synthetic division.
36.
37. Divide
38. Find the zeros of
by x + 3.
and state the multiplicity.
39. Write a polynomial function in standard form with zeros at 5, –4, and 1.
40. Find the zeros of
. Then graph the equation.
41. Use a graphing calculator to find the relative minimum, relative maximum, and zeros of
. If necessary, round to the nearest hundredth.
42. 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
. Graph the function. What is the maximum volume
for the domain
? Round to the nearest cubic foot.
43. Write 4x3 + 8x2 – 96x in factored form.
44. The table shows the number of llamas born on llama ranches worldwide since 1988. Find a cubic function to
model the data and use it to estimate the number of births in 1999.
Years since 1988
Llamas born (in thousands)
1
3
5
7
9
1.6
20
79.2
203.2
416
45. Write 4x2(–2x2 + 5x3) in standard form. Then classify it by degree and number of terms.
46. Write the polynomial
in standard form.
Essay
47. A model for the height of a toy rocket shot from a platform is
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.
, where x is the time in
Other
48. What are multiple zeros? Explain how you can tell if a function has multiple zeros.
49. Use division to prove that x = 3 is a real zero of
50. A polynomial equation with rational coefficients has the roots
additional roots and name them.
.
and
. Explain how to find two
Polynomial Review
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
ANS:
ANS:
ANS:
ANS:
C
A
D
D
SHORT ANSWER
5. ANS:
a.
b.
c.
The cubic model is a better fit.
6. ANS:
a.
y
b.
12
8
4
–4
4
8
x
–4
–8
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
7. ANS:
8.
9.
10.
11.
12.
a. height, x – 1; width, x – 3
b. height, 4 ft; width, 2 ft
ANS:
quintic binomial
ANS:
quintic trinomial
ANS:
; cubic trinomial
ANS:
0.0625
ANS:
13. ANS:
14. ANS:
15. ANS:
16. ANS:
17. ANS:
6 complex roots; 0, 2, 4, or 6 real roots; possible rational roots:
18. ANS:
7 complex roots; 1, 3, 5, or 7 real roots; possible rational roots: ±1, ±5
19. ANS:
4 complex roots; 0, 2 or 4 real roots
20. ANS:
21. ANS:
22. ANS:
23. ANS:
24. ANS:
3 ± 5i, –4
25. ANS:
–4, –2, –1, 1, 2, 4
26. ANS:
3, –3, 5, –5
27. ANS:
4 feet
28. ANS:
7
 ,
5
29. ANS:
30. ANS:
31. ANS:
9.3%
32. ANS:
15 in. by 20 in. by 44 in.
33. ANS:
0, –2, 0.38
34. ANS:
no solution
35. ANS:
4
36. ANS:
37. ANS:
, R –93
38. ANS:
–3, multiplicity 2; 5, multiplicity 6
39. ANS:
40. ANS:
0, 3, 2
y
6
4
2
–6
–4
–2
2
4
6
x
–2
–4
–6
KEY: Zero Product Property | polynomial function | zeros of a polynomial function | graphing
41. ANS:
relative minimum: (0.36, –62.24), relative maximum: (–3.69, 37.79),
zeros: x = –5, –2, 2
42. ANS:
145 ft3
43. ANS:
4x(x – 4)(x + 6)
44. ANS:
; 741,600 llamas
45. ANS:
20x5 – 8x4; quintic binomial
46. ANS:
ESSAY
47. ANS:
[4] a.
y
300
200
100
–2
2
4
6
8
10 x
–100
–200
b.
c.
d.
OTHER
48. ANS:
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.
about 336 feet; The height is the maximum value of the function.
If a linear factor of a polynomial is repeated, then the zero is repeated and the function has multiple zeros. To
determine whether a function has a multiple zero, factor the polynomial. If a factor is repeated in the factored
expression, then it is a multiple zero.
49. ANS:
÷ (x – 3) =
50. ANS:
By the Irrational Root Theorem, if
its conjugate
with no remainder, so x = 3 is a real zero of the function.
is a root, then its conjugate –
is also a root. Two additional roots are –
and
is also a root. If
.
is a root, then