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2.09 Polynomials of higher orders

Worksheet
Power functions
1

Do the following graphed functions have an even or odd power?

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
d
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
e
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
2

Consider the function y = x^{2}.

a

Complete the following table of values:

x- 3- 2- 10123
y
b

Using the points in the table, plot the curve on a cartesian plane.

c

Are the y-values ever negative?

d

Write down the equation of the axis of symmetry.

e

What is the minimum y-value?

f

For every y-value greater than 0, how many corresponding x values are there?

3

Consider the function f \left( x \right) = - x^{2}.

a

Does the graph rise or fall to the right?

b

Does the graph rise or fall to the left?

4

Consider the functions f(x) = - x^{4} and g(x) = - x^{6}.

a

Graph f(x) = - x^{4} and g(x) = - x^{6} on the same set of axes.

b

Which of the above functions has the narrowest graph?

5

Consider the functions f(x) = x^{3} and g(x) = x^{5}.

a

Graph f(x) = x^{3} and g(x) = x^{5}.

b

How would the graph of y = x^{7} differ to the graph of f(x) = x^{3} and g(x) = x^{5} ?

6

Consider the function y = x^{7}.

a

As x approaches infinity, what happens to the corresponding y-values?

b

As x approaches negative infinity, what happens to the corresponding y-values?

c

Sketch the general shape of y = x^{7}.

d

Sketch the general shape of y = - x^{7}.

Polynomials
7

Sketch the graph of the function f \left( x \right) = x^{5} - x^{3}.

8

Consider the function y = x^{4} - x^{2}.

a

Determine the leading coefficient of the polynomial function.

b

Is the degree of the polynomial odd or even?

c

Does y = x^{4} - x^{2} rise or fall to the left?

d

Does y = x^{4} - x^{2} rise or fall to the right?

e

Sketch the graph of y = x^{4} - x^{2}.

9

Consider the function which has intercepts \left( - 4 , 0\right), \left(2, 0\right) and \left(0, 3\right).

a

What is the lowest degree of a polynomial that goes through these points?

b

Sketch the graph of the quadratic function that has the given intercepts.

10

Consider the function f \left( x \right) = - 6 x^{2} - 4 x + 5 which is concave down.

a

State the coordinates of the y-intercept of the function.

b

How many x-intercepts does the function have?

11

Match each function to its correct graph:

a
f \left( x \right) = 2 x^{4} - x^{2} + 2
b
f \left( x \right) = - x^{3} + x^{2} - 3 x + 4
c
f \left( x \right) = x^{5} + \dfrac{x}{10} - 3
A
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
8
9
y
B
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
C
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
D
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
12

The graph of y = P \left(x\right) is shown. Sketch the graph of y = - P \left(x\right).

-5
-4
-3
-2
-1
1
2
3
4
5
x
-40
-30
-20
-10
10
20
30
40
y
13

Consider the function y = - x^{5} + x^{3}.

a

What does y approach as x \to -\infty?

b

What does y approach as x \to \infty?

c

What are the x-intercepts of f \left( x \right)?

d

What is the y-intercept of the function?

e

Complete the following table of values:

x- 2- 1012
y
f

Sketch the graph of the function.

14

Consider the function f \left( x \right) = x^{7} - 9 x^{3} - 2.

a

What is the maximum number of real zeros that the function can have?

b

What is the maximum number of x-intercepts that the graph of the function can have?

c

What is the maximum number of turning points that the graph of the function can have?

15

Consider the function f \left( x \right) = - 4 x^{5} - 6 x^{2} - 2 x - 9.

a

What is the maximum number of real zeros that the function can have?

b

What is the maximum number of x-intercepts that the graph of the function can have?

c

What is the maximum number of turning points that the graph of the function can have?

Factored form of polynomials
16

Sketch the graph of the function f \left( x \right) = x \left(x + 3\right) \left(x - 3\right).

17

For each of the following functions:

i

Find the x-intercepts.

ii

Find the y-intercept.

iii

Sketch the graph.

a
y = - \left(x - 1\right)^{2} \left(x + 2\right)
b
y = \left(x - 1\right) \left(x - 2\right) \left(x + 4\right) \left(x + 5\right)
c
y = - \left(x + 1\right) \left(x + 3\right) \left(x + 4\right) \left(x - 4\right)
d
y = \left(x - 2\right)^{2} \left(x + 3\right) \left(x - 1\right)
e
y = - \left(x - 3\right)^{2} \left(x + 1\right) \left(x - 2\right)
f
y = - \left(x - 3\right)^{2} \left(x + 2\right)^{2}
g
y = \left(x + 2\right)^{3} \left(x - 2\right)
18

Consider the function y = - \left(x - 1\right)^{2} \left(x^{2} - 9\right).

a

Find the x-intercepts.

b

Find the y-intercept.

c

Does the graph have y-axis symmetry, origin symmetry, or neither?

d

Sketch the graph of the function.

19

Consider the function y = - x \left(x + 2\right) \left(x - 2\right) \left(x - 3\right).

a

What does y approach as x \to -\infty?

b

What does y approach as x \to \infty?

c

What are the x-intercepts?

d

What is the y-intercept?

e

Complete the following table of values:

x- 2- 10123
y
f

Sketch the graph of the function.

Expanded form of polynomials
20

Consider the function y = x^{3} - 6 x^{2} + 3 x + 10.

a

Does the graph rise or fall to the right?

b

Does the graph rise or fall to the left?

c

Express the equation in factorised form.

d

Find the x-intercepts.

e

Find the y-intercept.

f

Sketch the graph of the function.

21

Consider the function y = x^{4} - 4 x^{2}.

a

State the leading coefficient.

b

Does the function rise or fall to the left?

c

Does the function rise or fall to the right?

d

Express the equation in factorised form.

e

State the x-intercepts.

f

Find the y-intercept.

g

Sketch the graph.

22

Consider the function y = - 4 x^{3} + 11 x^{2} - 5 x - 2.

a

What does y approach as x \to -\infty?

b

What does y approach as x \to \infty?

c

What are the possible integer or rational roots?

d

Complete a table of values to test for the roots of the polynomial.

e

What is the y-intercept of the function?

f

Sketch the graph.

23

Consider the function y = 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x.

a

What does y approach as x \to -\infty?

b

What does y approach as x \to \infty?

c

The polynomial has a linear factor of x. Write the polynomial as a product of x and a cubic polynomial.

d

Hence, write down one of the roots of the polynomial.

e

Hence, find the rational and integer roots of the cubic factor.

f

Sketch the graph of y = 2 x^{4} + 3 x^{3} - 2 x^{2} - 3 x.

Zeros of a polynomial
24

Solve the following equations:

a
\left(x^{2} - 9\right) \left(x^{2} + 12 x + 36\right) = 0
b
81 x^{4} - 121 x^{2} = 0
c
x^{5} + 5 x^{4} - 24 x^{3} = 0
25

Consider the function f \left( x \right) = x^{4} - 7 x^{3} + 12 x^{2} + 4 x - 16.

a

Is 5 a zero of the function?

b

Is 4 a zero of the function?

c

Is - 1 a zero of the function?

26

Consider the function f \left( x \right) = \left(x + 1\right)^{2} \left(x - 6\right) \left(x + 2\right)^{5}.

a

What are the zeros of the function?

b

State the multiplicity of each zero by filling in the table:

ZeroMultiplicity
- 1
6
- 2
27

Consider the function f \left( x \right) = - 3 \left(x - 6\right)^{4} \left(x + 2\right)^{5} x^{3}.

a

What are the zeros of the function?

b

State the multiplicity of each zero by filling in the table:

ZeroMultiplicity
6
- 2
0
28

The polynomial P \left( x \right) = x^{2} + k x + 8 has a zero at x = 4.

a

Find the other zero of P \left( x \right).

b

Find the value of k.

29

Consider the polynomial P \left( x \right) = x^{7} - 7 x^{4} - 7 x^{3} + 12. Use the constant term to write down all the possible rational zeros.

30

Consider the polynomial f \left( x \right) = x^{4} + 2 x^{3} + 42 x^{2} + 12 x + 42.

a

What are the possible rational zeros of f \left( x \right)?

b

What are the actual rational zeros of this polynomial?

31

The polynomial P \left( x \right) = x^{3} - 12 x - 9 has a zero at x = - 3. Find the other zeros of P \left( x \right).

32

The polynomial P \left( x \right) = x^{4} - 23 x^{2} + 112 has zeros at x = 4 and x = - 4. Find the other zeros of P \left( x \right).

33

Consider the polynomial:

f \left( x \right) = x^{4} + 4 x^{3} - 9 x^{2} - 26 x - 30

a

What are the possible rational zeros of f \left( x \right)?

b

What are the actual rational zeros of f \left( x \right)?

34

Consider the polynomial P \left( x \right) = x^{3} + 4 x^{2} + x - 6

a

Write down all the possible rational zeros.

b

The graph of P \left( x \right) is shown.

State which of the possible zeros listed in the previous part are actual zeros of P \left( x \right).

c

Factorise P \left( x \right).

-3
-2
-1
1
2
3
x
-8
-6
-4
-2
2
4
6
8
y
35

Consider the polynomial P \left( x \right) = x^{3} + 4 x^{2} + x - 6

a

Write down all the possible rational zeros.

b

The graph of P \left( x \right) is shown.

State which of the possible zeros listed in the previous part are actual zeros of P \left( x \right).

c

Factorise P \left( x \right).

-3
-2
-1
1
2
3
x
-6
-4
-2
2
4
6
y
36

The cubic equation x^{3} + k x^{2} + m x + 12 = 0 has a double root at x = 2.

a

Find the other root of the equation.

b

Find the value of k.

c

Find the value of m.

d

Sketch the graph of the function.

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Outcomes

ACMMM014

recognise features of the graphs of y=x^n for n∈N, n=−1 and n=½, including shape, and behaviour as x→∞ and x→−∞

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