Name: _______________________________
Date: ________________________
Chapter 2 Prerequisite Skills
Evaluate Functions
1. Given P(x) = x4– 3x2 + 5x – 11, evaluate.
a) P(–2)
b) P(3)
c) P(–1)
⎛1⎞
d) P ⎜ ⎟
⎝4⎠
Simplify Expressions
2. Expand and simplify.
a) (x3 – 2x2 – 3x + 4)(2x – 1) + 3
b) (3x3 + x2 – 4x – 2)(x + 3) – 5
c) (x –
6 )(x +
6)
d) (x – 2 7 )(x + 2 7 )
e) (x + 3 –
5 )(x + 3 +
5)
Factor Expressions
3. Factor fully.
a) x2 – 49
b) 64a2 – 121b2
c) 3m2 – 75n2
d) 5x4 – 5
4. Factor each trinomial.
a) b2 – 2b – 15
b) m2 – 9m +18
c) 2a2 – 5a – 12
d) 3x2 – 17x + 10
e) 6x2 – 5x – 4
…BLM 2–1. .
Determine Equations of Quadratic
Functions
7. Determine an equation for the quadratic
function, with the given zeros, that passes
through the given point.
a) zeros: –2 and 5; point (1, –12)
b) zeros: 6 and 0; point (–2, –32)
5
1
c) zeros: – and ; point (–1, 48)
2
3
Determine Intervals From Graphs
8. For the graph of each polynomial
function,
i) identify the x-intercepts
ii) write the interval(s) for which the
graph is above the x-axis and the
interval(s) for which the graph is
below the x-axis.
a)
b)
Solve Quadratic Equations
5. Solve by factoring.
a) x2 – 2x – 35 = 0
b) 5x2 – 16x +3 = 0
c) 18a2 – 50 = 0
d) 6x2 – 33x = 18
e) 10x2 – 7 = 9x
6. Use the quadratic formula to solve.
Round answers to one decimal place.
a) 3x2 – 5x + 1 = 0
b) 2x2 + 6x – 7 = 0
c) 7x + 13 = 4x2
Advanced Functions 12: Teacher’s Resource
BLM 2–1 Prerequisite Skills
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
2.1 The Remainder Theorem
1. a) Divide 2x3 – 3x2 + x – 6 by x + 2.
Express the result in quotient form.
b) Identify any restrictions on the
variable.
c) Write the corresponding statement that
can be used to check the division.
d) Verify your answer.
2. Perform each division. Express the result
in quotient form. Identify any restrictions
on the variable.
a) x3 + 2x2 – 5x + 3 divided by x + 2
b) 4x3 + 3x – 4 divided by 2x + 1
c) 6x3 – 9x4 + 6x – 5 divided by 3x – 2
d) 8x3 – 10x2 – 21 divided by x – 3
3. Determine the remainder R so that each
statement is true.
a) (3x – 2)(2x + 1) + R = 6x2 – x – 7
b) (x + 5)(2x – 1)(x + 4) + R =
2x3 + 17x2 + 31x – 10
4. The area, in square centimetres, of the
base of a square-based box is
4x2 – 12x + 9. Determine possible
dimensions of the box if the volume, in
cubic centimetres, is
4x3 – 28x2 + 57x – 36.
Date: ________________________
…BLM 2–2. .
7. For what value of k will the polynomial
f(x) = x3 + 2x2 + kx + 5 have the same
remainder when it is divided by x + 1 and
x – 2?
8. Use the remainder theorem to determine
the remainder when x4 + x3 – 3x + 6 is
divided by 3x – 2.
9. a) Use the remainder theorem to
determine the remainder when
4x3 +2x2 – 6x + 1 is divided by 2x – 1.
b) Verify your answer in part a) using
long division.
c) Use Technology Verify your answer
in part a) using technology.
10. a) Determine the remainder when
8x3 + 2x2 – x is divided by 2x + 1.
b) Factor 8x3 + 2x2 – x fully.
11. When the polynomial mx3 + 5x2 – nx – 1
is divided by x – 1, the remainder is 2.
When the same polynomial is divided by
x + 3, the remainder is 10. Determine the
values of m and n.
5. Use the remainder theorem to determine
the remainder for each division.
a) x3 – 3x2 + 5x – 2 divided by x – 4
b) 3x3 + x2 – 4x + 10 divided by x + 3
c) x4 + 2x3 – 3x + 2 divided by x + 2
6. a) Determine the value of c such that
when P(x) = 2x3 – cx2 + 4x – 7 is
divided by x – 2, the remainder is –3.
b) Use Technology Verify your answer
in part a) using a computer algebra
system.
Advanced Functions 12: Teacher’s Resource
BLM 2–2 Section 2.1 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
2.2 The Factor Theorem
1. Determine if x – 2 is a factor of each
binomial.
a) 3x3 – x2 + 2x
b) x3 + 2x2 – 16
c) 2x3 – 3x2 + x – 6
2. List the values that could be zeros of each
polynomial. Then, factor the polynomial.
a) x3 – 2x2 – x + 2
b) x3 – 7x – 6
c) x3 + 5x2 – 2x – 24
3. Factor each polynomial by grouping
terms.
a) x3 + 2x2 – 9x – 18
b) 2x3 + 5x2 – 8x – 20
c) x3 – 3x2 – 25x + 75
d) 3x3 – 5x2 – 27x + 45
4. Determine the values that could be zeros
of each polynomial. Then, factor the
polynomial.
a) x3 + x2 – 10x + 8
b) 2x3 + 5x2 + x – 2
c) 2x3 + 3x2 – 5x – 6
d) 3x3 – 16x2 + 23x – 6
5. Factor each polynomial.
a) x3 + 5x2 – x – 5
b) x3 – 7x + 6
c) x3 – 3x2 – 4x + 12
d) x4 + 4x3 – x2 – 16x – 12
e) x4 – 3x3 – 14x2 + 48x – 32
6. Use Technology Factor each polynomial.
a) 2x3 + 3x2 – 11x – 6
b) 4x3 – 9x2 – 10x + 3
c) 5x3 – 12x2 – 36x + 16
7. Determine the value of k so that x – 3 is a
factor of x3 – 2x2 + kx – 6.
…BLM 2–3. .
9. A carpenter is building a rectangular
storage shed whose volume, V, in cubic
metres, can be modelled by
V(x) = 4x3 – 36x2 + 107x – 105.
a) Determine the possible dimensions of
the shed, in terms of x, in metres, that
result in the volume in part a).
b) What are the dimensions of the shed
when x = 5.2?
10. Factor each polynomial.
a) 2x3 + 11x2 + 2x – 15
b) 3x3 + 8x2 + 3x – 2
c) 5x3 – 17x2 + 16x – 4
d) 4x3 + 5x2 – 23x – 6
11. Factor each polynomial.
a) 8x3 – 125
8
b) 64x3 +
27
c) 216x3 + y3
d) 27 – t6
1 3
e) 125x6 –
y
64
f) 8x6 + 343y12
12. Factor each polynomial by letting t = x2.
a) 16x4 – 17x2 + 1
b) 9x4 – 61x2 + 100
13. Determine a polynomial function P(x)
that satisfies the following set of
⎛2⎞
conditions: P(2) = P ⎜ ⎟ = P(– 4) = 0
⎝3⎠
and P(3) = –7.
14. Factor.
3x5 – 2x4 – 22x3 – 4x2 + 19x + 6
8. Determine the value of k so that 2x + 5 is
a factor of 4x3 – kx2 – 6x + 10.
Advanced Functions 12: Teacher’s Resource
BLM 2–3 Section 2.2 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
2.3 Polynomial Equations
1. Solve.
a) x3 – 4x2 + 3x = 0
b) 2x3 + x2 – 18x – 9 = 0
c) 3x3 – 2x2 – 12x + 8 = 0
2. Solve.
a) 2x3 – 3x2 – 11x + 6 = 0
b) x3 – x2 – 17x – 15 = 0
c) 8x3 – 6x2 – 3x + 1 = 0
3. Use the graphs to determine the roots of
the corresponding polynomial equations.
The roots are all integral values.
a) Window variables: x ∈ [–7, 7],
y ∈ [–20, 20], Yscl = 5
b) Window variables: x ∈ [–7, 7],
y ∈ [–20, 40], Yscl = 5
c) Window variables: x ∈ [–7, 5],
y ∈ [–10, 10]
…BLM 2–5. .
5. Determine the x-intercepts of the graph of
each polynomial function.
a) f(x) = x3 – 1
b) g(x) = x3 + 3x2 + 4x + 12
c) h(x) = x5 – 9x3 + 8x2 – 72
d) y = x4 – 25x2 + 144
6. Solve.
a) x3 – 2x2 – 5x + 6 = 0
b) x4 – x3 – 10x2 – 8x = 0
c) 5x5 – 80x = 0
d) 2x3 + 3x2 – 23x – 12 = 0
e) 3x3 + 7x2 = 4
f) x4 – 26x2 + 25 = 0
7. Use Technology Solve. Round answers
to two decimal places.
a) x3 + 3x2 + 7x – 1 = 0
b) x4 – 6x3 + 2x2 = 3
c) 3x3 – 4x = 2x2 – 8
d) 4x4 – 6x2 – 2x – 4 = 0
8. Use Technology A rectangular water
tank in an aquarium has width 2x – 5,
length x + 4, and height 2x – 3, with all
the dimensions in metres. If the volume
of the tank is 110 m3, use technology to
solve a polynomial equation in order to
determine the approximate dimensions of
the tank, to two decimal places.
9. The length of a child’s square-based
jewellery box is 5 cm more than its
height. The box has a capacity of
500 cm3. Solve a polynomial equation to
determine the dimensions of the box.
4. Determine the real roots of each
polynomial equation.
a) (x – 1)(x2 + 2x + 4) = 0
b) (x2 + 5x + 10)(x2 – 25) = 0
c) (4x2 – 64)(5x2 + 25) = 0
d) (x3 – 1)(3x2 – 27) = 0
Advanced Functions 12: Teacher’s Resource
BLM 2–5 Section 2.3 Practice
10. Find all real and complex solutions to
3x3 + 12x2 + 13x – 28 = 0.
11. Determine a polynomial equation of
degree 3 with roots x = 5 and x = 7 ± 2i.
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
2.4 Families of Polynomial Functions
…BLM 2–6. .
(page 1)
1. The zeros of a quadratic function are –3
and 5.
a) Determine an equation for the family
of functions with these zeros.
b) Write equations for two functions with
these zeros.
c) Determine an equation for the member
of the family that passes through the
point (–1, 6).
2. Examine the following functions. Which
function does not belong to the same
family? Explain.
A y = 4(2x + 1)(x – 5)(x + 7)
B y = 4(x – 5)(2x + 1)(x + 7)
C y = –4(x – 5)(x + 7)(2x + 1)
D y = 4(x + 7)(2x – 1)(x – 5)
3. The graphs of three polynomial functions
are given. Which graph represents a
function that does not belong to the same
family as the other two? Explain.
A Window variables: x ∈ [–7, 7],
y ∈ [–20, 20], Yscl = 2
B Window variables: x ∈ [–7, 7],
y ∈ [–10, 10]
C Window variables: x ∈ [–7, 7],
y ∈ [–10, 5]
5. Which of the following polynomial
functions belong to the same families?
Explain.
A y = –0.8(x – 4)(x + 1)(x + 3)
2
B y = – (x – 1)(x + 3)(x + 4)
3
C y = 0.8(x – 4)(x + 3)(x + 1)
D y = 0.5(x + 1)(x – 4)(x + 3)
E y = –2(x – 1)(x + 4)(x + 3)
F y = 3(x + 3)(x – 1)(x + 4)
6. a) Write an equation for a family of
functions with each set of zeros.
i) –5, 2, 7
ii) –6, –2, 3
iii) –4, –1, 2, 5
b) Determine an equation for the member
of the family that passes through the
point (1, 8) for each equation in
part a).
7. a) Determine an equation for the family
of cubic functions with zeros –2, 2,
and 5.
b) Write equations for two functions that
belong to the family in part a).
c) Determine an equation for the member
of the family whose graph has a
y-intercept of 10.
d) Sketch a graph of the functions in
parts b) and c).
8. a) Determine an equation for the family
of quartic functions with zeros –4, –1,
0, and 3.
b) Write equations for two functions that
belong to the family in part a).
c) Determine an equation for the member
of the family whose graph passes
through the point (2, 36).
d) Sketch a graph of the functions in
parts b) and c).
4. Determine an equation for the function
that corresponds to each graph in
question 3.
Advanced Functions 12: Teacher’s Resource
BLM 2–6 Section 2.4 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
…BLM 2–6. .
(page 2)
9. a) Determine an equation for the family
3
of cubic functions with zeros − , 1,
2
5
and .
2
b) Determine an equation for the member
of the family whose graph passes
through the point (–1, –28).
c) Sketch a graph of the function in
part b).
10. a) Determine an equation, in simplified
form, for the family of cubic functions
with zeros 2 and 4 ± 3 .
b) Determine an equation for the member
of the family whose graph passes
through the point (1, –18).
11. Determine an equation for the cubic
function represented by this graph.
13. An open-top box is to be constructed
from a square piece of cardboard that has
sides measuring 30 cm each. It is
constructed by cutting congruent squares
from the corners and then folding up the
sides.
a) Express the volume of the
square-based box as a function of x.
b) Write an equation to represent a box
with a volume that is
i) one-half the volume of the box
represented by the function in
part a)
ii) three times the volume of the box
represented by the function in
part a)
c) How are the equations in part b)
related to the one in part a)?
d) Sketch graphs of the functions from
parts a) and b) on the same coordinate
grid.
e) Determine possible dimensions of the
box that has a volume of 1728 cm3.
14. a) Write an equation for a family of odd
functions with three x-intercepts, two
5
5
of which are − and .
2
2
b) Determine an equation, in simplified
form, for the member of the family in
part a) that passes through the point
(–3, 66).
c) Determine an equation, in simplified
form, for the member of the family in
part b) that is a reflection in the x-axis.
d) Is the function in part c) an odd
function? Explain.
12. a) Determine an equation, in simplified
form, for the family of quartic
functions with zeros 1 (order 2) and
–3 ± 5 .
b) Determine an equation for the member
of the family in part a) whose graph
has a y-intercept of –12.
Advanced Functions 12: Teacher’s Resource
BLM 2–6 Section 2.4 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
2.5 Solve Inequalities Using Technology
1. Write inequalities for the values of x
shown.
a)
b)
…BLM 2–7. .
(page 1)
5. For each graph write
i) the x-intercepts
ii) the intervals of x for which the graph
is positive.
iii) the intervals of x for which the graph
is negative.
a) Window variables: x ∈ [–6, 6],
y ∈ [–5, 15]
c)
d)
2. Write the intervals into which the x-axis
is divided by each set of x-intercepts of a
polynomial function.
a) –7, –1
b) 3, 4
c) –2, 6, 0
3. Describe what the solution to each
inequality indicates about the graph of
y = f(x).
a) f(x) > 0 when –3 < x < –1 or x > 2
b) f(x) ≤ 0 when –2 ≤ x ≤ 0 or
0≤ x ≤ 2
4. Sketch a graph of a quartic polynomial
function y = f(x) such that f(x) > 0 when
–2.5 < x < –0.5 or 1 < x < 3 and f(x) < 0
when x < –2.5 or –0.5 < x < 1 or x > 3.
b) Window variables: x ∈ [–8, 6],
y ∈ [–10, 10]
c) Window variables: x ∈ [–6, 6],
y ∈ [–10, 10]
d) Window variables: x ∈ [–6, 6],
y ∈ [–10, 10]
6. Solve each polynomial inequality by
graphing the polynomial function.
a) x2 – 2x – 8 ≤ 0
b) x2 + 7x + 6 > 0
c) x3 + x2 – 16x – 16 ≥ 0
d) x3 – 2x2 – 5x + 6 < 0
e) x3 – 4x2 – 11x + 30 ≤ 0
Advanced Functions 12: Teacher’s Resource
BLM 2–7 Section 2.5 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
…BLM 2–7. .
(page 2)
7. Solve each polynomial inequality. Use a
computer algebra system, if available.
a) 2x2 – 5x – 3 < 0
b) 4x2 – 28x + 45 ≥ 0
c) x3 – 4x2 + x + 6 > 0
d) x3 + x2 – 9x – 9 ≤ 0
e) x3 – 6x2 – x + 30 ≥ 0
8. Use Technology Solve each polynomial
inequality by first finding the
approximate zeros of the related
polynomial function. Round answers to
two decimal places.
a) 2x2 – 5x + 1 ≥ 0
b) 2x3 + x2 – 3x – 1 < 0
c) –4x3 – 2x + 5 > 0
d) x3 + 2x2 – 4x – 6 ≤ 0
e) 3x4 – 5x2 – 4x + 5 < 0
10. The height, h, in metres, of a golf ball
t seconds after it is hit can be modelled
by the function h(t) = –4.9t2 + 32t + 0.2.
When is the height of the ball less than
10 m? Round to two decimal paces.
11. The solutions given correspond to an
inequality involving a quartic function.
Write a possible quartic polynomial
inequality.
5
3
5
x < – or
< x < or x > 7
2
2
2
12. Use Technology Solve. Round answers
to two decimal places.
3x4 + 8x3 + x2 – 10 ≤ 10x4 + 3x3 – 8x – 4
9. Solve. Round answers to one decimal
place.
a) 3x3 – 2x2 – 12x – 12 > 0
b) 2x3 + x2 + 3x – 2 < 0
c) –x3 + 10x – 5 ≤ 0
d) –2x4 + 6x3 – x2 + 3x – 10 ≥ 0
Advanced Functions 12: Teacher’s Resource
BLM 2–7 Section 2.5 Practice
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
2.6 Solve Factorable Polynomial Inequalities Algebraically
1. Solve each inequality. Show each
solution on a number line.
a) 3x – 2 > 7
b) 3 – x ≥ 5
c) 5x – 11 > 2x + 1
d) 4(2 – 3x) ≤ 2x – 6
2. Solve by considering all cases. Show
each solution on a number line.
a) (x + 2)(x – 3) ≥ 0
b) (2x + 1)(x – 2) < 0
3. Solve using intervals. Show each solution
on a number line.
a) (x + 4)(3x – 5) > 0
b) (3x + 2)(x – 1) ≤ 0
4. Solve.
a) (x + 2)(x – 4)(x – 6) ≥ 0
b) (3x + 5)(2x – 1)(x – 3) ≤ 0
c) (1 – x)(–2x + 3)(x – 2) > 0
d) (2 – 3x)(x + 1)(3x – 2) < 0
…BLM 2–8. .
7. Solve.
a) x2 – 2x – 24 < 0
b) x3 + 6x2 + 11x + 6 ≥ 0
c) –2x3 + 7x2 – 2x – 3 > 0
d) –x3 + 5x2 – 2x – 8 ≤ 0
8. A certain type of candle is packaged in
boxes that measure 36 cm by 15 cm by
8 cm. The candle company that produced
the above packaging has now designed
shorter candles. A smaller box will be
created by decreasing each dimension of
the larger box by the same length. The
volume of the smaller box will be at the
most 930 cm3. What are the maximum
dimensions of the smaller box?
9. Solve using intervals.
3x4 + 10x3 + 12 ≤ 2x5 + 15x2 + 8x
5. Solve by considering all cases. Show
each solution on a number line.
a) x2 + 3x – 10 < 0
b) x2 + 10x + 21 ≥ 0
c) 2x3 + 3x2 – 3x – 2 ≤ 0
d) 3x3 – x2 – 12x + 4 > 0
6. Solve using intervals.
a) x3 – 2x2 – 5x + 6 ≥ 0
b) –x3 + 5x2 – 2x – 8 > 0
c) 3x3 – 5x2 + 2x < 0
d) x4 – 13x2 – 12x ≤ 0
Advanced Functions 12: Teacher’s Resource
BLM 2–8 Section 2.6 Practice
Copyright®2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Chapter 2 Review
2.1 The Remainder Theorem
1. i) Use the remainder theorem to
determine the remainder for each
division.
ii) Perform each division. Express the
result in quotient form. Identify all
restrictions on the variable.
a) x3 + 4x2 – 3 divided by x – 2
b) 3x3 – 5x2 + 2x – 6 divided by x – 5
c) 2x4 – 3x3 – 4x2 + 5x – 15 divided by
2x + 1
2. a) Determine the value of k such that
when f(x) = 3x5 – 4x3 + kx2 – 1 is
divided by x + 2, the remainder is –5.
b) Use Technology Verify your answer
in part a) using technology.
3. For what value of m will the polynomial
P(x) = 2x3 + mx2 – 4x + 1 have the same
remainder when it is divided by x + 2 and
by x – 3?
2.2 The Factor Theorem
4. List the values that could be zeros of each
polynomial. Then, factor the polynomial.
a) x3 + x2 – 10x + 8
b) 2x3 + 7x2 + 7x + 2
c) 3x4 + x3 – 14x2 – 4x + 8
5. Factor each polynomial.
a) x3 – 3x2 – 9x + 27
b) 4x3 + 4x2 – 25x – 25
c) 9x3 + 18x2 – 4x – 8
6. Determine the value of b such that x + 4
is a factor of 2x3 – 4x2 + bx – 8.
7. Determine the value of k such that 3x – 2
is a factor of x3 + kx2 – 5x + 3.
8. A rectangular box of crackers has a
volume, in cubic centimetres, that can be
modelled by the function
V(x) = x3 – 33x2 + 300x – 800.
a) Determine the dimensions of the box
in terms of x.
b) What are the possible dimensions of
the box when x = 25?
Advanced Functions 12: Teacher’s Resource
BLM 2–9 Chapter 2 Review
Date: ________________________
…BLM 2–9. .
(page 1)
2.3 Polynomial Equations
9. Use the graph to determine the roots of
the corresponding polynomial equation.
Window variables: x ∈ [–6, 6],
y ∈ [–25, 25], Yscl = 5
10. Determine the x-intercepts of each
polynomial function.
a) y = 27x3 – 64
b) f(x) = x3 – 2x2 + 16x – 32
c) g(x) = x4 – 29x2 + 100
11. Determine the real roots of each
polynomial equation.
a) (x2 – 3x – 10)(2x2 + 8) = 0
b) (5x2 – 125)(3x3 – 81) = 0
12. Use Technology Solve. Round answers
to two decimal places.
a) 5x3 + 2x2 + 3x + 10 = 0
b) 5x – 2x3 = 18 – 9x2
c) 4x4 + 3x3 + 2x – 1 = 0
13. Use Technology A small doll house has
dimensions such that the width is 6 cm
less than the height and the length is 3 cm
less than 1.5 times the height.
a) Write an equation for the volume of
the house.
b) Find the possible dimensions of the
house, to two decimal places, if the
volume is 8500 cm3.
2.4 Families of Polynomial Functions
14. a) Determine an equation for the family
1
of cubic functions with zeros – , 2,
2
and 6.
b) Write equations for two functions that
belong to the family in part a).
c) Determine an equation for the member
whose graph passes through the point
(–1, 42).
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
…BLM 2–9. .
(page 2)
15. a) Determine an equation, in simplified
form, for the family of cubic functions
with zeros –3 and 1 ± 6 .
b) Determine an equation for the member
of the family whose y-intercept is –10.
2.5 Solving Inequalities Using Technology
16. Use Technology Solve. Round answers
to two decimal places, if necessary.
a) x3 – 5x2 + 4x – 3 ≥ 0
b) –3x3 + 4x > 0
c) 2x4 + 5x3 – x2 + x – 3 ≤ 0
d) 4x5 + 7x3 – 2x + 10 < 0
2.6 Solve Factorable Polynomial
Inequalities Algebraically
18. Solve each inequality. Show the solution
on a number line.
a) (4x + 5)(x + 2) ≥ 0
b) (3x – 1)(2x + 5)(3 – x) ≤ 0
c) (4x2 – 9)(x2 + 6x + 9) > 0
19. Solve.
a) 2x2 – x – 15 < 0
b) –x3 – x2 + 9x + 9 > 0
c) x4 – 4x3 – 21x2 + 100x – 100 ≤ 0
17. Sketch a graph of a cubic polynomial
function y = f(x) such that f(x) < 0 when
x < –5 or –3 < x < 2 and f(x) > 0 when
–5 < x < –3 or x > 2.
Advanced Functions 12: Teacher’s Resource
BLM 2–9 Chapter 2 Review
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Chapter 2 Test
For questions 1 to 3, select the best answer.
1. Which of the following is not a factor of
2x3 – x2 – 18x + 9?
Ax+3
Bx–3
C 2x + 1
D 2x – 1
2. Which statement is false for
P(x) = –2x3 + 11x2 – 19x + 10?
A P(x) = (x + 1)(–2x2 + 13x – 32) + 42
B 2x – 5 is a factor of P(x).
C When P(x) is divided by x – 2, the
remainder is 10.
D x – 2 is a factor of P(x).
3. The values that could be zeros for the
polynomial x3 – 2x2 – 19x + 20 are
A ± 1, ± 4, ± 5
B ± 1, ± 2, ± 4, ± 5, ± 10, ± 20
C ± 1, ± 2, ± 4, ± 5
D ± 1, ± 2, ± 4, ± 5, ± 10
Date: ________________________
…BLM 2–11. .
(page 1)
7. Use the graph to determine the roots of
the corresponding polynomial equation.
Window variables: x ∈ [–6, 6],
y ∈ [–20, 10], Yscl = 2
8. Solve by factoring.
a) 2x3 – x2 – 6x = 0
b) x3 – 2x2 – 5x + 6 = 0
c) 2x4 + 3x3 – 7x2 – 12x – 4 = 0
9. Determine an equation for the cubic
function represented by this graph.
4. a) Divide 3x3 – x2 – 1 by x + 2. Express
the result in quotient form.
b) Identify any restrictions on the
variable.
c) Write the corresponding statement that
can be used to check the division.
5. a) Determine the value of k such that
when P(x) = x4 – 2x2 + kx – 4 is
divided by x + 3, the remainder is 2.
b) Determine the remainder when P(x) is
divided by 2x – 1.
c) Verify your answer in part b) using
long division.
6. Factor.
a) x3 – 125y3
b) x3 – 4x2 – 9x + 36
c) x3 + 4x2 + x – 6
d) 3x3 + 8x2 + 3x – 2
e) x4 – 4x3 – x2 + 16x – 12
Advanced Functions 12: Teacher’s Resource
BLM 2–11 Chapter 2 Test
10. Determine an equation, in simplified
form, for the family of quartic functions
with zeros 1 ± 3 and 2 ± 5 .
11. Use Technology Solve. Round answers
to one decimal place.
a) 3x3 – 6x2 + x – 6 ≥ 0
b) 2x4 – x2 – 2 < 5x – 3x3
12. Solve by factoring.
a) 4x2 – 64 ≥ 0
b) –x3 + 2x2 + 8x < 0
c) x4 – 3x3 – 3x2 + 7x + 6 > 0
Copyright © 2008 McGraw-Hill Ryerson Limited
Name: _______________________________
Date: ________________________
…BLM 2–11. .
(page 2)
13. An open-top box is to be constructed
from a rectangular piece of cardboard
measuring 52 cm by 36 cm. The box is
created by cutting congruent corners and
then folding up the sides.
a) Express the volume of the box as a
function of x.
b) Use your function from part a) to
determine the value(s) of x, to two
decimal places, that will result in a
volume that is greater than 3024 cm3.
c) Determine the dimensions of the box
for the volume given in part b).
Advanced Functions 12: Teacher’s Resource
BLM 2–11 Chapter 2 Test
Copyright © 2008 McGraw-Hill Ryerson Limited
Chapter 2 Practice Masters Answers
Prerequisite Skills
1. a) –17
b) 58
c) –18
2543
d) –
256
2. a) 2x4 – 5x3 – 4x2 + 11x – 1
b) 3x4 + 10x3 – x2 – 14x – 11
c) x2 – 6
d) x2 – 28
e) x2 +6x + 4
3. a) (x – 7)(x + 7)
b) (8a – 11b)(8a + 11b)
c) 3(m – 5n)(m + 5n)
d) 5(x – 1)(x + 1)(x2 + 1)
4. a) (b – 5)(b + 3)
b) (m – 3)(m – 6)
c) (2a + 3)(a – 4)
d) (3x – 2)(x – 5)
e) (2x + 1)(3x – 4)
1
or x = 3
5. a) x = –5 or x = 7
b) x =
5
1
5
5
c) x = − or x =
d) x = – or x = 6
3
3
2
1
7
e) x = – or x =
2
5
6. a) x 0.2 or x 1.4
b) x –3.9 or x 0.9
c) x –1.1 or x 2.9
7. a) y = x2 – 3x – 10 b) y = –2x2 + 12x
c) y = –24x2 – 52x + 20
8. a) i) –1 and 3
ii) above the x-axis: –1 < x < 3; below
the x-axis: x < –1 or x > 3
b) i) –2, 1, and 3
ii) above the x-axis: x < –2 or 1 < x < 3;
below the x-axis: –2 < x < 1 or x > 3
2.1 The Remainder Theorem
− 36
1. a) 2x2 – 7x + 15 +
x+2
b) x ≠ –2
c) (2x2 – 7x + 15)(x + 2) – 36
x 3 + 2x 2 − 5x + 3
13
2. a)
,
= x2 – 5 +
x+2
x+2
x ≠ –2
−6
4 x 3 + 3x − 4
b)
= 2x2 – x + 2 +
,
2x + 1
2x + 1
1
x≠–
2
Advanced Functions 12: Teacher’s Resource
BLM 2–13 Chapter 2 Practice Masters Answers
…BLM 2–13. .
(page 1)
− 9x 4 + 6x 3 + 6x − 5
=
3x − 2
2
−1
, x≠
–3x3 + 2 +
3x − 2
3
3
2
8 x − 10 x − 21
d)
=
x−3
105
, x≠3
8x2 +14x + 42 +
x−3
3. a) –5
b) 10
4. (2x – 3) cm by (2x – 3) cm by (x – 4) cm
One possible answer: 7 cm by 7 cm by 1 cm
5. a) 34
b) –50
c) 8
6. a) 5
7. –5
40
8. 4
81
9. a) –1
10. a) 0
b) x(2x + 1)(4x – 1)
5
11
11. m = , n =
3
3
c)
2.2 The Factor Theorem
1. a) not a factor
b) factor
c) factor
2. a) ± 1, ± 2; (x – 2)(x – 1)(x + 1)
b) ± 1, ± 2, ± 3, ± 6; (x + 2)(x + 1)(x – 3)
c) ± 1, ± 2, ± 3, ± 4, ± 6, ± 8, ± 12,
± 24; (x + 4)(x + 3)(x – 2)
3. a) (x + 2)(x – 3)(x + 3)
b) (2x + 5)(x – 2)(x + 2)
c) (x – 3)(x – 5)(x + 5)
d) (3x – 5) (x – 3)(x + 3)
4. a) ± 1, ± 2, ± 4, ± 8; (x – 1)(x – 2)(x + 4)
1
b) ± 1, ± 2, ± ; (2x – 1)(x + 1)(x + 2)
2
1
3
c) ± 1, ± 2, ± 3, ± 6, ± , ± ;
2
2
(x + 1)(x + 2)(2x – 3)
1
2
d) ± 1, ± 2, ± 3, ± 6, ± , ± ;
3
3
(3x – 1)(x – 2)(x – 3)
5. a) (x + 1)(x – 1)(x + 5)
b) (x – 1)(x – 2)(x + 3)
c) (x – 3)(x + 2)(x – 2)
d) (x + 1)(x + 2)(x – 2)(x + 3)
e) (x – 1)(x – 2)(x + 4)(x – 4)
Copyright © 2008 McGraw-Hill Ryerson Limited
Chapter 2 Practice Masters Answers
6. a) (2x + 1)(x – 2)(x + 3)
b) (x + 1)(4x – 1)(x – 3)
c) (x – 4)(x + 2)(5x – 2)
7. –1
8. –6
9. a) (x – 3)(2x – 5)(2x – 7)
b) 2.2 m by 5.4 m by 3.4 m
10. a) (2x + 3)(x – 1)(x + 5)
b) (3x – 1)(x + 1)(x + 2)
c) (5x – 2)(x – 1)(x – 2)
d) (4x + 1)(x + 3)(x – 2)
11. a) (2x – 5)(4x2 + 10x + 25)
2 ⎞⎛
8
4⎞
⎛
b) ⎜ 4 x + ⎟⎜16 x 2 − x + ⎟
3 ⎠⎝
3
9⎠
⎝
2
2
c) (6x + y)(36x – 6xy + y )
d) (3 – t2)(9 + 3t2 + t4)
1 ⎞⎛
5
1 2⎞
⎛
e) ⎜ 5 x 2 − y ⎟⎜ 25 x 4 + x 2 y +
y ⎟
4 ⎠⎝
4
16 ⎠
⎝
f) (2x2 + 7y4)(4x4 – 14x2y4 + 49y8)
12. a) (4x – 1)(4x +1)(x – 1)(x + 1)
b) (3x – 5)(3x + 5)(x – 2)(x + 2)
1
13. y = – (x – 2)(3x – 2)(x + 4)
7
14. (x – 1)(x +2)(x – 3)(3x + 1)(x + 1)
2.3 Polynomial Equations
1. a) x = 0 or x = 1 or x = 3
1
b) x = – or x = 3 or x = –3
2
2
c) x =
or x = 2 or x = –2
3
1
or x = 3
2. a) x = –2 or x =
2
b) x = –3 or x = –1 or x = 5
1
1
or x = 1
c) x = – or x =
2
4
3. a) x = –2 or x = 1 or x = 3
b) x = –5 or x = –3 or x = 2
c) x = –4 or x = –2 or x = 0
4. a) x = 1
b) x = 5 or x = –5
c) x = 4 or x = –4
d) x = 3 or x = –3 or x = 1
5. a) 1
b) –3
c) –2, –3, 3
d) –4, –3, 3, 4
Advanced Functions 12: Teacher’s Resource
BLM 2–13 Chapter 2 Practice Masters Answers
…BLM 2–13. .
(page 2)
6. a) x = –2 or x = 1 or x = 3
b) x = –2 or x = –1 or x = 0 or x = 4
c) x = 0 or x = –2 or x = 2
1
d) x = –4 or x = – or x = 3
2
2
e) x = –2 or x = –1 or x =
3
f) x = –5 or x = –1 or x = 1 or x = 5
7. a) x 0.13
b) x –0.68 or x 5.66
c) x –1.47
d) x –1.31 or x 1.51
8. 2.86 m by 7.93 m by 4.86 m
9. 10 cm by 10 cm by 5 cm
− 15 ± i 111
10. x = 1 or x =
6
11. x3 – 19x2 + 123x – 265 = 0
2.4 Families of Polynomial Functions
1. a) y = k(x + 3)(x – 5), k ∈ , k ≠ 0
b) Answers may vary.
1
c) y = – (x + 3)(x – 5)
2
2. D (has different zeros)
3. C (has different zeros)
4. a) y = –2x(x + 2)(x – 3)
1
b) y = x(x + 2)(x – 3)
2
1
c) y = x(x – 4)(x + 2)
2
5. A, C, and D (zeros are –3, –1, 4); B, E,
and F (zeros are –4,–3, 1)
6. a) i) y = k(x + 5)(x – 2)(x – 7)
ii) y = k(x + 6)(x + 2)(x – 3)
iii) y = k(x + 4)(x + 1)(x – 2)(x – 5)
2
b) i) y = (x + 5)(x – 2)(x – 7)
9
4
ii) y = – (x + 6)(x + 2)(x – 3)
21
1
iii) y = (x + 4)(x + 1)(x – 2)(x – 5)
5
7. a) y = k(x + 2)(x – 2)(x – 5)
b) Answers may vary.
1
c) y = (x + 2)(x – 2)(x – 5)
2
d) Answers may vary.
Copyright © 2008 McGraw-Hill Ryerson Limited
Chapter 2 Practice Masters Answers
8. a) y = k(x)(x + 4)(x + 1)(x – 3)
b) Answers may vary.
c) y = –x(x + 4)(x + 1)(x – 3)
d) Answers may vary.
9. a) y = k(2x + 3)(x – 1)(2x – 5)
b) y = –2(2x + 3)(x – 1)(2x – 5)
c)
10. a) y = k(x3 – 10x2 + 29x – 26)
b) y = 3(x3 – 10x2 + 29x – 26)
3
11. y = (x + 2)(x – 1)(x – 4)
4
12. a) y = k (x4 + 4x3 – 7x2 – 2x + 4)
b) y = –3(x4 + 4x3 – 7x2 – 2x + 4)
13. a) V(x) = x(30 – 2x)(30 – 2x)
b) i) V(x) = x(15 – x)(30 – 2x)
ii) V(x) = 3x(30 – 2x)(30 – 2x)
c) They all have the same zeros.
d)
e) 3 cm by 24 cm by 24 cm or 7.3 cm by
15.4 cm by 15.4 cm
14. a) Answers may vary. Sample answer:
y = k(x)(2x – 5)(2x + 5)
b) y = –8x3 + 50x
c) y = 8x3 – 50x
d) It is an odd function, since
f(–x)= –f(x).
Advanced Functions 12: Teacher’s Resource
BLM 2–13 Chapter 2 Practice Masters Answers
…BLM 2–13. .
(page 3)
2.5 Solve Inequalities Using Technology
1. a) –4 ≤ x ≤ 2
b) –6 < x < –1
c) –2 < x ≤ 5
d) 1 ≤ x < 3
2. a) x < –7, –7 < x < –1, x > –1
b) x < 3, 3 < x < 4, x > 4
c) x < –2, –2 < x < 0, 0 < x < 6, x > 6
3. a) intervals where f(x) is above the x-axis
b) intervals where f(x) is on or below the
x-axis
4.
5. a) i) –3, 4
ii) –3 < x < 4
iii) x < –3, x > 4
b) i) –5, 2
ii) x < –5, x > 2
iii) –5 < x < 2
c) i) –2, 1, 5
ii) –2 < x < 1, x > 5
iii) x < –2, 1 < x < 5
d) i) –3, –1, 2, 4
ii) –3 < x < –1, 2 < x < 4
iii) x < –3, –1 < x < 2, x > 4
6. a) –2 ≤ x ≤ 4
b) x < –6 or x > –1
c) –4 ≤ x ≤ –1 or x ≥ 4
d) x < –2 or 1 < x < 3
e) x ≤ –3 or 2 ≤ x ≤ 5
7. a) –0.5 < x < 3
b) x ≤ 2.5 or x ≥ 4.5
c) –1 < x < 2 or x > 3
d) x ≤ –3 or –1 ≤ x ≤ 3
e) –2 ≤ x ≤ 3 or x ≥ 5
8. a) x ≤ 0.22 or x ≥ 2.28
b) x < –1.34 or –0.32 < x < 1.16
c) x < 0.92
d) x ≤ –2.66 or –1.21 ≤ x ≤ 1.87
e) 0.77 < x < 1.31
9. a) x > 2.7
b) x < 0.5
c) –3.4 ≤ x ≤ 0.5 or x ≥ 2.9
d) 1.3 ≤ x ≤ 2.8
10. from 0 s to 0.32 s and between 6.21 s
and 6.54 s
11. Answers may vary. Sample answer:
8x4 – 68x3 + 34x2 + 425x – 525 > 0
12. x ≤ 0.68 or x ≥ 1.14
Copyright © 2008 McGraw-Hill Ryerson Limited
Chapter 2 Practice Masters Answers
2.6 Solve Factorable Polynomial
Inequalities Algebraically
1. a) x > 3
b) x ≤ –2
c) x > 4
d) x ≥ 1
2. a) x ≤ –2 or x ≥ 3
1
b) − < x < 2
2
5
2
3. a) x < –4 or x >
b) – ≤ x ≤ 1
3
3
4. a) –2 ≤ x ≤ 4 or x ≥ 6
5
1
b) x ≤ – or
≤ x≤ 3
3
2
3
c) 1 < x < or x > 2
2
2
2
d) –1 < x <
or x >
3
3
5. a) –5 < x < 2
b) x ≤ –7 or x ≥ –3
1
c) x ≤ –2 or −
≤ x ≤ 1
2
1
or x > 2
d) –2 < x <
3
6. a) –2 ≤ x ≤ 1 or x ≥ 3
b) x < –1 or 2 < x < 4
2
c) x < 0 or < x < 1
3
d) –3 ≤ x ≤ –1 or 0 ≤ x ≤ 4
7. a) –4 < x < 6
b) –3 ≤ x ≤ –2 or x ≥ –1
1
c) x < − or 1 < x < 3
2
d) –1 ≤ x ≤ 2 or x ≥ 4
8. 31 cm by 10 cm by 3 cm
3
9. –2 ≤ x ≤ –1 or 1 ≤ x ≤
or x ≥ 2
2
Chapter 2 Review
1. a) i) 21
x3 + 4 x2 − 3
21
= x 2 + 6 x + 12 +
,
ii)
x−2
x−2
x≠2
b) i) 254
3x 3 − 5 x 2 + 2 x − 6
=
ii)
x −5
254
3x 2 + 10 x + 52 +
,x≠5
x−5
Advanced Functions 12: Teacher’s Resource
BLM 2–13 Chapter 2 Practice Masters Answers
…BLM 2–13. .
(page 4)
2 x 4 − 3x 3 − 4 x 2 + 5 x − 15
=,
2x +1
−18
1
x3 − 2 x2 − x + 3 +
,x≠ −
2x + 1
2
2. a) 15
3. –10
4. a) ± 1, ± 2, ± 4, ± 8; (x – 1)(x – 2)(x + 4)
1
b) ± 1, ± 2, ± ; (2x + 1)(x + 1)(x + 2)
2
2
1
4
c) ± 1, ± 2, ± 4, ± 8, ± , ± , ± ,
3
3
3
8
± ; (x + 1)(x + 2)(3x – 2)(x – 2)
3
5. a) (x – 3)2(x + 3)
b) (x + 1)(2x – 5)(2x + 5)
c) (x + 2)(3x – 2)(3x + 2)
6. –50
1
7.
12
8. a) (x – 20) cm by (x – 5) cm by (x – 8) cm
b) 5 cm by 20 cm by 17 cm
9. x = –5 or x = –1 or x = 2
4
10. a)
b) 2
c) –5, –2, 2, 5
3
11. a) x = –2 or x = 5
b) x = 3 or x = 5 or x = –5
12. a) x –1.23
b) x –1.46 or x 1.34 or x 4.62
c) x –1.22 or x 0.38
13. a) V(x) = x(x – 6)(1.5x – 3)
b) 20.68 cm by 14.68 cm by 28.02 cm
14. a) y = k(2x + 1)(x – 2)(x – 6)
b) Answers may vary.
c) y = –2(2x + 1)(x – 2)(x – 6)
15. a) y = k(x3 + x2 – 11x – 15)
2
b) y = (x3 + x2 – 11x – 15)
3
16. a) x ≥ 4.22
b) x < –1.15 or 0 < x < 1.15
c) –2.81 ≤ x ≤ 0.76
d) x < –1.02
17.
c) i) –18 ii)
Copyright © 2008 McGraw-Hill Ryerson Limited
Chapter 2 Practice Masters Answers
18. a) x ≤ –2 or x ≥ –
…BLM 2–13. .
(page 5)
5
4
5
1
≤ x ≤ or x ≥ 3
2
3
3
3
c) x < –3 or –3 < x < – or x >
2
2
5
19. a) – < x < 3
2
b) x < –3 or –1 < x < 3
c) –5 ≤ x ≤ 5
b) –
Chapter 2 Test
1. C
2. C
3. B
3x 3 − x 2 − 1
− 29
4. a)
= 3x2 – 7x +14 +
x+2
x+2
b) x ≠ –2
c) (3x2 – 7x +14)(x + 2) – 29
81
5. a) 19
b)
16
2
6. a) (x – 5y)(x + 5xy + 25y2)
b) (x – 3)(x + 3)(x – 4)
c) (x – 1)(x + 2) (x + 3)
d) (3x – 1)(x + 2)(x + 1)
e) (x – 1)(x + 2)(x – 3)(x – 2)
7. x = –4 or x = –1 or x = 1 or x = 3
3
8. a) x = – or x = 0 or x = 2
2
b) x = –2 or x = 1 or x = 3
1
c) x = –2 or x = –1 or x = – or x = 2
2
9. y = –(x + 4)(x + 1)(x – 2)
10. y = x4 – 6x3 + 5x2 + 10x + 2
11. a) x ≥ 2.2
b) –0.5 < x <1.2
12. a) x ≤ – 4 or x ≥ 4
b) –2 < x < 0 or x > 4
c) x < –1 or –1 < x < 2 or x > 3
13. a) V(x) = x(36 – 2x)(52 – 2x)
b) 1.96 < x < 13.52
c) 1.96 cm by 32.08 cm by 48.08 cm or
13.52 cm by 8.96 cm by 24.96 cm
Advanced Functions 12: Teacher’s Resource
BLM 2–13 Chapter 2 Practice Masters Answers
Copyright © 2008 McGraw-Hill Ryerson Limited