MHF4U
Characteristics of Polynomials Quiz 1
NAME: ________________
Please do not use graphing calculators for this assessment.
1. Consider a polynomial relation of the form y=−x5 + a x 3 +bx +c , where a , … , c ∈ R. Fill in the requested
information in the table below.
End behaviour Domain
Type of range
Number of
Number of
(circle one):
zeros
turning points
As x →−∞ ,
Describe
(list all
(list all
poly
{ y ∈ R}
y→
possibilities):
possibilities):
K/A
State
{ y ∈ R∨ y ≥ k }
Features
As x → ∞ ,
{ y ∈ R∨ y ≤ k }
y→
2. a) Check off all relations that could represent polynomial functions. Note: each graph shows all the turning
points for its function.
3 y−2 x
=5
2
□
□
□
□
□
□
□
b) Determine whether the table of values below represents a polynomial function.
If so, state its degree; if not, explain why not. Show supporting work.
Compare
poly and
non-poly
fn’s
y=3 x 3 +2 x+ x−1
x 2+ y 2=1
y=2x +4
x -2 -1
y -16 -1
0
0
1
1
2
16
K/A
Classify
3. Match the graphs to their equations. Label each graph by writing the capital
letter of the matching function (A, B, C, or D).
A(x )=x (x +1)2 ( x−1)3
B( x)=x 2(x +1)3 (x−1)
T
C (x)=x 3 (x +1)( x−1)2
Determine Eqn
from Graph
D( x )=x ( x−1)2(x +1)3
Match
4. a) Sketch a polynomial function y=f ( x )that satisfies all of the following conditions:




There is a zero of order 2 at x=2
There is a zero of order 1 at x=1
There is a zero of order 3 at x=−3
As x → ± ∞ , y →−∞
b) Determine an equation in factored form of the
function from part (a) if it passes through the point
(0, 54).
A
Determine
poly fn from
conditions
Sketch
K
C
Equation
Neatness and
Organization
5. For each function given below, determine whether the function has even symmetry, odd symmetry, or
neither. Show your reasoning.
a)
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
b) f (x)=x2 (x 3−1)( x 3 +1)
x
1 2 3 4 5 6 7 8 9
Determine
Symmetry
K
Understanding
concepts,
procedures
C
Clarity and
Conventions
MHF4U
Characteristics of Polynomials Quiz 1
NAME: ________________
Please do not use graphing calculators for this assessment.
1. Consider a polynomial relation of the form y=−x 4 +ax 3 +b x 2+ cx+ d , where a , … , d ∈ R. Fill in the
requested information in the table below.
End behaviour Domain
Type of range
Number of
Number of
(circle one):
zeros
turning points
As x →−∞ ,
(list all
(list all
{ y ∈ R}
y→
possibilities):
possibilities):
{
y ∈ R∨ y ≥ k }
As x → ∞ ,
{ y ∈ R∨ y ≤ k }
y→
Describe
poly
K/A
State
Features
2. a) Circle all relations that could represent polynomial functions. Note: each graph shows all the turning
points for its function.
□
□
□
□
□
□
□
x= y 2 +1
y=3 x 3 +1 x 1−1 x−1
3 y−2 x
=5
2
y=3 x +2 x +1 x
b) Determine whether the table of values below represents a polynomial function.
If so, state its degree; if not, explain why not. Show supporting work.
Compare
poly and
non-poly
fn’s
x -3 -2 -1 0 1
y 34 10 0 -2 -2
K/A
Classify
3. Match the graphs to their equations. Label each graph by writing the capital
letter of the matching function (A, B, C, or D).
A(x )=x (x +1)2 (x−1)3
B( x)=x 2(x +1)3 (x−1)
T
C (x)=x 3 ( x +1)(x−1)2
Determine Eqn
from Graph
D( x )=x (x−1)2( x +1)3
Match
4. a) Sketch a polynomial function y=f ( x )that satisfies all of the following conditions:




There is a zero of order 2 at x=−2
There is a zero of order 1 at x=1
There is a zero of order 3 at x=3
As x → ± ∞ , y →−∞
b) Determine an equation in factored form of the
function from part (a) if it passes through the point
(0,−54).
A
Determine
poly fn from
conditions
Sketch
K
C
Equation
Neatness and
Organization
5. For each function given below, determine whether the function has even symmetry, odd symmetry, or
neither. Show your reasoning.
a)
9
8
7
6
5
4
3
2
1
-9 -8 -7 -6 -5 -4 -3 -2 -1-1
-2
-3
-4
-5
-6
-7
-8
-9
b) f (x)=x2 (x 3−1)( x 3 +1)
x
1 2 3 4 5 6 7 8 9
Determine
Symmetry
K
Understanding
concepts,
procedures
C
Clarity and
Conventions
MHF4U
Characteristics of Polynomials Quiz 2
NAME: ________________
Please do not use graphing calculators for this assessment.
1. Consider a polynomial relation of the form y=an xn + an−1 x n−1 +…+a 2 x 2+ a1 x+ a0, where a i ∈ R . Answer
each question below:
Suppose as
Can the domain be
Can the range be
How many xHow many
x → ± ∞ , y →∓ ∞. {x ∈ R∨x ≥ 0 }? If yes,
turning points
{ y ∈ R∨ y ≥ 50 } ? If yes, intercepts are
Under which
possible if n is are possible if
state the conditions; if
state the conditions; if
conditions would
odd?
n is even?
not, explain why not.
not, explain why not.
this be true?
Describe
poly
K/A
State
Features
2
.
State 2 features of the following
graph that indicate that it is not a
polynomial relation.
Compare
poly and
non-poly fn’s
Check off the equations that
represent polynomial
relations.
□
□
□
□
□
K/A
y=2x + x1 +10
Determine whether the table of values
below represents a polynomial relation of
degree 5. Show your reasoning.
x 4
6
8 10 12 14 16
y 7
7
6
2 -3 -6 0
√ y=x 2 +1
y+ x
=1
2
y+ 2
=1
x
xy=1
Classify
3. a) Sketch a polynomial function y=f ( x )of degree 4 that satisfies all of the following conditions:



There is a zero of order 2 at x=−2
f ( 1 ± √2 ) =0
f ( 1 ) =3
K/A
Compare
poly and
non-poly
fn’s
Classify
b) Determine an equation in factored form of the
function from part (a).
A
Determine
poly fn from
conditions
Sketch
K
C
Equation
Neatness and
Organization
4. For each function given below, determine whether the function has even symmetry, odd symmetry, or
neither. Show your reasoning.
a)
b) f (x)=x3 (x 2−x )11
Determine
Symmetry
K
Understanding
concepts,
procedures
C
Clarity and
Conventions
MHF4U
Characteristics of Polynomials Quiz 2
NAME: ________________
Please do not use graphing calculators for this assessment.
1. Consider a polynomial relation of the form y=an xn + an−1 x n−1 +…+a 2 x 2+ a1 x+ a0, where a i ∈ R . Answer
each question below:
Suppose as
Can the domain be
Can the range be
How many xHow many
x → ± ∞ , y →−∞. {x ∈ R∨x ≥ 0 }? If yes,
turning points
{ y ∈ R } ? If yes, state the intercepts are
Under which
possible if n is are possible if
state the conditions; if
conditions; if not,
conditions would
even?
n is odd?
not, explain why not.
explain why not.
this be true?
Describe
poly
K/A
State
Features
2
.
State 2 features of the following
graph that indicate that it is not a
polynomial relation.
Compare
poly and
non-poly fn’s
Check off the equations that
represent polynomial
relations.
□
□
□
□
□
K/A
y=2x + x1 +10
Determine whether the table of values
below represents a polynomial relation of
degree 5. Show your reasoning.
x 4
6
8 10 12 14 16
y 7
7
6
2 -3 -6 0
√ y=x 2 +1
y+ x
=1
2
y+ 2
=1
x
xy=1
Classify
3. a) Sketch a polynomial function y=f ( x )of degree 4 that satisfies all of the following conditions:



There is a zero of order 2 at x=−2
f ( 1 ± √ 3 )=0
f ( 1 ) =9
K/A
Compare
poly and
non-poly
fn’s
Classify
b) Determine an equation in factored form of the
function from part (a).
A
Determine
poly fn from
conditions
Sketch
K
C
Equation
Neatness and
Organization
4. For each function given below, determine whether the function has even symmetry, odd symmetry, or
neither. Show your reasoning.
a)
b) f (x)=x3 (x 2−x )10
Determine
Symmetry
K
Understanding
concepts,
procedures
C
Clarity and
Conventions
MHF4U0
Polynomial Algebra Quiz 1
NAME:_________
1. Determine the remainder when x 160 −2 x 3 +5 is divided by x +1.
K
Understand
remainder
theorem
Calculations
2. Use the most efficient method to factor each of the following polynomials. Clearly show your procedures.
a) 4 x 4 +12 x 2 +9
b) 40 x 4 + 96 x3 +5 x +12
3. Determine the number of rational and irrational zeros for the function f ( x )=4 x 4 + x 2−3 x +1.
4. Suppose two rectangular prisms with the dimensions shown below have equal volume. Determine this
volume. Make sure to check that your solution makes sense in this context. Show your strategy and
calculations.
x
K
Work with
polynomial
equations
5.
Factoring
strategies
A
Solve
polynomial
equations
C
Organization
Interpret results
and
conventions
Determine if x=−2 satisfies the inequality
2 x−3 ≥ 4 x+ 6. Show your strategy.
C (x)
6.
7.
T
Use the graph on the right to determine the values of x for
which P ( x )<C ( x ). Show your solution set along a number line
and write the solution using an inequality or interval notation.
Demonstrate a strategy for solving each inequality below algebraically.
a) 2 x+5> 7 x −3
b) (2−3 x ) ( x−1 )2 (−x 2−2 x +2)≤ 0
Work with
polynomial
inequalities
K
Check
inequality
solution
A
Solve
inequalities
graphically
K
Solve
inequalities
algebraically
C
Organization
and
conventions
P( x )
MHF4U0
Polynomial Algebra Quiz 1
NAME:_________
1. Determine the remainder when x 161−2 x 3 +5 is divided by x +1.
K
Understand
remainder
theorem
Calculations
2. Use the most efficient method to factor each of the following polynomials. Clearly show your procedures.
a) 4 x 4−12 x 2+ 9
b) 40 x 4 −96 x 3+ 5 x −12
3. Determine the number of rational and irrational zeros for the function f ( x )=4 x 4 + x 2 +3 x+1.
4. Suppose two rectangular prisms with the dimensions shown below have equal volume. Determine this
volume. Make sure to check that your solution makes sense in this context. Show your strategy and
calculations.
x
K
Work with
polynomial
equations
5.
Factoring
strategies
A
Solve
polynomial
equations
C
Organization
Interpret results
and
conventions
Determine if x=−2 satisfies the inequality
2 x−3 ≥ 4 x+ 6. Show your strategy.
C (x)
6.
7.
T
Use the graph on the right to determine the values of x for
which P ( x ) ≥ C( x ). Show your solution set along a number line
and write the solution using an inequality or interval notation.
Demonstrate a strategy for solving each inequality below algebraically.
a) 2 x+5> 7 x −3
b) (2−3 x ) ( x−1 )2 (−x 2−2 x +2)≥ 0
Work with
polynomial
inequalities
K
Check
inequality
solution
A
Solve
inequalities
graphically
K
Solve
inequalities
algebraically
C
Organization
and
conventions
P( x )
MHF4U0
Polynomial Algebra Quiz 2
NAME:_________
1. Determine the value of k if the remainder when x 4 +k x 3 +5 is divided by 2 x+1 is equal to
95
.
16
K
Understand
remainder
theorem
Calculations
2. Factor each polynomial below fully without repeating any strategies. Clearly show your procedures.
a) 64 x 6−16 x 3 +1
b) 9 x 4 +6 x 3 +10 x2 +6 x +1
3. The equation h ( d )=0.1 d(d−20)2, 0 ≤ d ≤ 20 models the height of a roller coaster, h, with respect to the
horizontal distance from the starting point, d (all measurements are in meters). The coaster operator wishes
to place a camera on the coaster track 100 m above ground to snap pictures of the riders on their way up on
the first slope. Determine the horizontal distance from the starting point at which the camera should be
placed. Make sure to check that your solution makes sense in this context. Show your strategy and
calculations.
K
Work with
polynomial
equations
Factoring
strategies
A
Solve
polynomial
equations
T
C
Organization
Interpret results
and
conventions
MHF4U0
Polynomial Algebra Quiz 2
NAME:_________
1. Determine the value of k if the remainder when x 4 +k x 3 +5 is divided by 3 x+ 1 is equal to
379
.
81
K
Understand
remainder
theorem
Calculations
2. Factor each polynomial below fully without repeating any strategies. Clearly show your procedures.
a) 9 x 4 −6 x3 +28 x 2−18 x+ 3
b)
x 6 +54 x 3+ 729
3. The equation h ( d )=0.05 d (d−3 0)2, 0 ≤ d ≤ 3 8 models the height of a roller coaster, h, with respect to the
horizontal distance from the starting point, d (all measurements are in meters). The coaster operator wishes
to place a camera on the coaster track 100 m above ground to snap pictures of the riders on their way up on
the second slope. Determine the horizontal distance from the starting point at which the camera should be
placed. Make sure to check that your solution makes sense in this context. Show your strategy and
calculations.
K
Work with
polynomial
equations
Factoring
strategies
A
Solve
polynomial
equations
T
C
Organization
Interpret results
and
conventions