Chapter 1 Review

Name: ______________________
Class: _________________
Date: _________
Chapter 1 Review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. What is the remainder when x 3 + 4 − 11x + 3x 2 is divided by 6 + x?
A. 70
____
B. −62
C . −38
2. Divide: (−4x − x 2 − 2 − x 3 ) ÷ (x − 3)
C . −x 2 − 4x − 16 R(–50)
D. −x 2 + 2x + 8 R46
A. −x 2 − 4x − 16
B. −x 2 + 2x + 8
____
3. Divide −3x 3 − 2x 2 + 4x + 3 by x + 3. Write the division statement.
A.
B.
C.
D.
____
−3x 3
−3x 3
−3x 3
−3x 3
− 2x 2
− 2x 2
− 2x 2
− 2x 2
+ 4x + 3
+ 4x + 3
+ 4x + 3
+ 4x + 3
B. 9
2x 4
2x 4
2x 4
2x 4
− 4x 3
− 4x 3
− 4x 3
− 4x 3
− 7x 2
− 7x 2
− 7x 2
− 7x 2
− 27x − 36
− 27x − 36
− 27x − 36
− 27x − 36
= (x − 4)(2x 3
= (x − 4)(2x 3
= (x − 4)(2x 3
= (x − 4)(2x 3
B. 15
D. –28
− 4x 2 − 12x − 45)
− 12x 2 − 25x − 45)
+ 9x 2 + 9x + 4)
+ 4x 2 + 9x + 9)
C . –21
7. Which polynomial has x + 3 as a factor?
A. x 3 − 12x 2 + 37x
B. x 3 − 9x 2 − 12x + 37
____
C . 39
6. For the polynomial P(x) = −3x 2 − 4x − 5, what is the value of P(−2)?
A. –25
____
− 11x + 25)
− 11x + 25) – 48
+ 7x − 17)
+ 7x − 17) + 54
5. Divide −7x 2 − 27x − 36 + 2x 4 − 4x 3 by x − 4. Write the division statement.
A.
B.
C.
D.
____
= (x + 3)(−3x 2
= (x + 3)(−3x 2
= (x + 3)(−3x 2
= (x + 3)(−3x 2
4. When x 4 + x 3 − 12x 2 + 41x + q is divided by x + 5, the remainder is 0. What is the value of q?
A. 5
____
D. 46
C . x 3 − 9x 2 + x
D. x 3 − 9x 2 + x + 111
8. Which two binomials are factors of x 4 + 8x 3 + 7x 2 − 40x − 60?
A. x + 2 and x − 6
B. x − 2 and x − 6
C . x − 2 and x + 6
D. x + 2 and x + 6
1
D. –9
ID: A
Name: ______________________
____
ID: A
9. What value of k will ensure x − 3 is a factor of kx 4 + 9x 3 − 4x 2 − 5x − 30?
A. 2
B. –3
C. 3
D. –2
____ 10. The volume of a shipping box with the shapeof a rectangular prism can be expressed as the polynomial
2x 3 + 11x 2 + 17x + 6. Each dimension of the box can be expressed as a binomial. Which binomial could
represent one dimension of the box?
A. 2x + 1
B. x + 1
C . 2x + 3
D. x + 6
____ 11. The graph of a polynomial function of degree 4 is shown. Which statements are true?
i) The function has an even degree.
ii) The function has a zero of multiplicity 2.
iii) The equation of the function has a negative leading coefficient.
iv) The y-intercept is positive.
A. i, ii, iii
B. i, iii, iv
C . ii, iii, iv
2
D. i, ii, iv
Name: ______________________
ID: A
____ 12. Identify the graph that corresponds to the function f(x) = −x 2 − x + 2.
A.
C.
B.
D.
____ 13. Determine the zeros of the polynomial function f(x) = (x + 2) 4 (x − 5) . State the multiplicity of each
zero.
A.
B.
C.
D.
The
The
The
The
zero
zero
zero
zero
4 has multiplicity 2; the zero 1 has multiplicity –5.
4 has multiplicity –2; the zero 1 has multiplicity 5.
–2 has multiplicity 4; the zero 5 has multiplicity 1.
2 has multiplicity 4; the zero –5 has multiplicity 1.
____ 14. A carton of juice in the shape of a rectangular prism has dimensions 5.2 cm by 5.2 cm by 9.4 cm. The
manufacturer wants to design a carton with double the capacity by increasing each dimension by x
centimetres. Which equation could be used to determine the value of x?
A. 508.352 = (5.2 − x) 2 (9.4 − x)
B. 254.176 = (5.2 + x) 2 (9.4 + x)
C . 508.352 = (5.2 + x) 2 (9.4 + x)
D. 254.176 = (5.2 − x) 2 (9.4 − x)
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Name: ______________________
ID: A
____ 15. A piece of cardboard 30 cm long and 16 cm wide is used to make a gift box that has a top.
The diagram shows the net for the
box. The shaded parts are discarded.
The squares cut from the corners have
side length x centimetres.
Which polynomial function represents the volume of the box in terms of x?
A. V = 2x(15 − x)(8 − x)
B. V = 4x(15 − x)(8 − x)
C . V = x(15 − x)(16 − x)
D. V = x(30 − 2x)(16 − x)
Short Answer
1. When 2x 2 − 7x + 104 is divided by x − 1 using synthetic division, the result is: 2 − 5 99
Write the division statement.
2. Determine the quotient and remainder when the polynomial 3x 3 + 3x 2 − 30x + 15 is divided by x + 4.
3. Divide: (−5x 5 − 20x 4 − 25x 3 − 12x 2 − 5x + 40) ÷ (x + 2)
Write the quotient and the remainder.
4. Determine one binomial factor of x 4 − 2x 3 − 17x 2 + 18x − 40.
5. Factor: 2x 3 + 9x 2 − 11x − 30
4
Name: ______________________
ID: A
6. The zeros of the polynomial function f(x) = x 4 − 3x 3 − 7x 2 + 15x + 18 are –1, –2, and 3.
The zeros –1 and –2 have multiplicity 1. The zero 3 has multiplicity 2.
Sketch a graph of the function.
7. A manufacturer designs a cylindrical can with no top. The surface area of the can is 325 cm2 . The can
has base radius r centimetres and height h centimetres.
Write a polynomial function to model the capacity, C
cubic centimetres, of the can as a function of r.
Problem
1. A polynomial is divided by x + 2. The quotient is 5x 2 + 5x + 9 and the remainder is 3. What is the original
polynomial? Explain your work.
2. Is 3x − 1 a factor of 3x 3 − x 2 − 15x + 10? Justify your answer.
5
Name: ______________________
ID: A
3. Match each graph with its equation. Justify your choices.
a) f(x) = − 3x 5 − 10x 3 + 7x − 4
b) g(x) = 3x 3 + 10x 2 − 7x + 4
c) h(x) = 4x 4 − 10x 2 − x − 4
d) p(x) = − 4x 4 − 4x 3 − x − 4
i) Graph A
ii) Graph B
iii) Graph C
iv) Graph D
4. A quartic function has these characteristics: leading coefficient is negative; zero 2 has multiplicity 2,
each of the zeros −1 and −3 has multiplicity 1. Sketch a possible graph of the function. Justify your
thinking. Label the graph with its equation and label the y-intercept.
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Name: ______________________
ID: A
5. Use intercepts to sketch the graph of this polynomial function: f(x) = 2x 4 − 7x 3 − 2x 2 + 13x + 6
Explain your steps.
6. Sketch the graph of this polynomial function: f(x) = (x + 1) 2 (x − 1) 3
7. Cheryl and Gina are twins. They were born 3 years before their younger brother, Ben. This year, the
product of their three ages is 4809 greater than the sum of their ages. How old are the twins? Show your
work.
8. A silo is in the shape of a cylinder with an inverted cone attached to the bottom. The cylinder has a
height of 8 m and the height of the cone is equal to the radius of the silo. If the volume of the silo is 81 π
3
m , determine the length of the diameter.
7
Name: ______________________
ID: A
9. An airplane manufacturer can produce up to 15 planes per month. The profit made from the sale of
2
these planes can be modeled byP(x) = −0.2x + 4x − 3 whereP(x) is the profit in hundred thousand of
dollars per month and x is the number of planes made and sold. Based on this model, how many planes
should be made and sold to maximize the profit and what is the maximum profit?
10. A boardwalk that is x feet wide is built around a rectangular pond. The pond is 30 ft. wide and 40 ft.
long. The combined surface area of the pond and the boardwalk is 2000 ft2 . What is the width of the
boardwalk?
11. The width of a rectangular prism is w centimeters. The height is 2 cm less than the width. The length is
4 cm more than the width. If the magnitude of the volume of the prism is 8 times the measure of the
length, what are the dimensions of the prism?
12. Rectangular blocks of granite are to be cut and used to build the front entrance of a new hotel. The
3
2
volume, V, in cubic meters of each block can be modeeled by the function V(x) = 2x + 7x + 2x − 3,
where x is in meters.
a. What are the possible dimensions of the blocks in terms of x?
b. What are the possible dimensions of the blocks when x=1?
8
ID: A
Chapter 1 Review
Answer Section
MULTIPLE CHOICE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
C
C
D
A
D
D
D
D
D
A
D
A
C
C
A
SHORT ANSWER
1. 2x 2 − 7x + 104 = (x − 1)(2x − 5) + 99
2.
3.
4.
5.
3x 2 − 9x + 6 R(–9)
The quotient is −5x 4 − 10x 3 − 5x 2 − 2x − 1 and the remainder is 42.
x + 4 or x − 5
(x − 2)(x + 5)(2x + 3)
6.
1
ID: A
7. C(r) =
325r − πr 3
2
PROBLEM
1. The division statement has the form:
Original polynomial = (binomial divisor)(quotient polynomial) + remainder
The binomial divisor is: x + 2
The quotient polynomial is: 5x 2 + 5x + 9
The remainder is: 3
Multiply the quotient by the divisor, then add the remainder.
(x + 2)(5x 2 + 5x + 9) + 3
= 5x 3 + 5x 2 + 9x + 10x 2 + 10x + 18 + 3
= 5x 3 + 15x 2 + 19x + 21
So, the original polynomial is: 5x 3 + 15x 2 + 19x + 21
2. Use long division.
2
x −5
3
2
3x − 1 3x − x − 15x + 10
3
2
3x − x
0 − 15x + 10
−15x + 5
5
Since there is a remainder of 5, 3x − 1 is not a factor of 3x 3 − x 2 − 15x + 10.
3. a) The equation f(x) = − 3x 5 − 10x 3 + 7x − 4 represents a quintic function. Since the coefficient of x 5 is
negative, the graph rises to the left and falls to the right. So, f(x) = − 3x 5 − 10x 3 + 7x − 4 matches
Graph C.
b) The equation g(x) = 3x 3 + 10x 2 − 7x + 4 represents a cubic function. Since the coefficient of x 3 is
positive, the graph falls to the left and rises to the right. So, g(x) = 3x 3 + 10x 2 − 7x + 4 matches Graph
A.
c) The equation h(x) = 4x 4 − 10x 2 − x − 4 represents a quartic function. Since the coefficient of x 4 is
positive, the graph opens up. So, h(x) = 4x 4 − 10x 2 − x − 4 matches Graph D.
d) The equation p(x) = − 4x 4 − 4x 3 − x − 4 represents a quartic function. Since the coefficient of x 4 is
negative, the graph opens down. So, p(x) = − 4x 4 − 4x 3 − x − 4 matches Graph B.
2
ID: A
4. The zero 2 has multiplicity 2, so the graph just touches the x-axis at x = 2.
Each of the zeros −1 and −3 has multiplicity 1, so the graph crosses the x-axis at x = −1 and x = −3.
Since the function is quartic, there are no more zeros.
The leading coefficient is negative, so as x → −∞, the graph falls to the left, and as x → ∞, the graph
falls to the right. That is, the graph opens down.
A possible graph is:
A possible equation is:
2
y = − (x − 2 ) (x + 1 ) (x + 3 )
2
The y-intercept for this function is: − ( −2 ) ( 1 ) ( 3 ) = −12
3
ID: A
5. Factor the polynomial. Use the factor theorem.
Use mental math. When x = −1, f(−1) = 0
So, x + 1 is a factor of f(x) = 2x 4 − 7x 3 − 2x 2 + 13x + 6.
Divide to determine the other factor.
So, 2x 4 − 7x 3 − 2x 2 + 13x + 6 = (x + 1)(2x 3 − 9x 2 + 7x + 6)
Factor the cubic polynomial. Use the factor theorem.
Let g(x) = 2x 3 − 9x 2 + 7x + 6
When x = 2, g(2) = 0
So, x − 2 is a factor.
Divide to determine the other factor.
So, 2x 3 − 9x 2 + 7x + 6 = (x − 2)(2x 2 − 5x − 3)
Factor the trinomial: 2x 2 − 5x − 3 = (x − 3)(2x + 1)
So, 2x 4 − 7x 3 − 2x 2 + 13x + 6 = (x + 1)(x − 2)(x − 3)(2x + 1)
1
The zeros of the function are: –1, 2, 3, −
2
1
The x-intercepts of the graph are: –1, 2, 3, −
2
The equation has degree 4, so it is an even-degree polynomial function.
The leading coefficient is positive, so the graph opens up.
The constant term is 6, so the y-intercept is 6.
A graph of f(x) = 2x 4 − 7x 3 − 2x 2 + 13x + 6 is:
4
ID: A
6. To determine the zeros, solve f(x) = 0.
0 = (x + 1) 2 (x − 1) 3
The roots of the equation are x = −1 and x = 1.
So, the zeros of the function are –1 and 1.
The zero –1 has multiplicity 2.
The zero 1 has multiplicity 3.
So, the graph just touches the x-axis at x = −1 and crosses the x-axis at x = 1.
The equation has degree 5, so it is an odd-degree polynomial function.
The leading coefficient is positive, so as x → −∞, the graph falls and as x → ∞, the graph rises.
The y-intercept is: (1) 2 (−1) 3 = −1
Plot points at the intercepts, then draw a smooth curve that falls to the left and rises to the right.
f(x) = (x + 1) 2 (x − 1) 3
5
ID: A
7. Let Ben’s age in years to be x.
Then, in years, Cheryl’s age is x + 3 and Gina’s age is x + 3.
The sum of their ages is: x + (x + 3) + (x + 3) = 3x + 6
Product of ages − sum of ages = 4809
So, x(x + 3)(x + 3) − (3x + 6) = 4809
x 3 + 6x 2 + 6x − 4815 = 0
Use a graphing calculator to graph y = x 3 + 6x 2 + 6x − 4815.
The x-intercept is 15, which is Ben’s age.
So, the twins are 15 + 3, or 18 years old.
8. diameter is 6 m.
9. 10 planes , maximum profit of $1,700,000.00
10. 5 ft.
11. 4 cm by 2 cm by 8 cm
12. a. x+3, 2x-1, x+1
b. 4m by 1m by 2m
6