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4.08 Polynomials

Worksheet
Features of polynomials
1

Determine whether the following are polynomials:

a

A \left( x \right) = 4 x^{\frac{1}{4}} + 2 x^{5} + 2

b

3 x^{3} + \dfrac{2}{x^{7}} - 1

c
2x^3+9x-4
d
9x^4+6x^2+\sqrt{x}+10
e
5x^4 - 9x^3 + 2x^2 + 11x
f
2x^5 - 60x^2 - \dfrac{1}{x}
2

Consider the term 6 x^{7}.

a

State the coefficient of the term.

b

State the index.

3

For the following expressions:

i

State the number of terms.

ii

State the coefficients of the terms.

a

9 n^{7}

b

- 8 k^{7} - k

c

t + 4 t^{9} - t^{7}

d

x^3 - 6x^2 +8x - 5

4

Consider the polynomial P \left( x \right) = 5.

a

State the degree.

b

State the constant term.

5

For the following polynomials:

i

State the degree.

ii

State the leading coefficient.

iii

State the constant term.

a

P \left( x \right) = 5 x^{4} + 7 x^{2} + 5 x + 3

b

P \left( x \right) = x^{5} - 3 x^{4} + 6 x^3 - x^2 + 20x-11

c

P \left( x \right) = 4 - \dfrac{6}{7} x^{6}

d

P \left( x \right) = \dfrac{x^{6}}{2} + \dfrac{x^{5}}{10} + 6

6

State the degree of the following terms:

a

8

b

- 4 y^{3}

c

- 4^{8} y^{6}

7

State the numerical coefficient of:

a
3 k^{6}
b
- k
c
\dfrac{t}{3}
d
\dfrac{6 m}{7}
8

Consider the polynomial P \left(x\right) = 4 x^{4} + 2 x^{5} - 5.

a

Find P \left(0\right).

b

Find P \left(1\right).

c

Find P \left(-4\right).

9

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

a

Find P\left(-1\right).

b

Find P\left(\sqrt{2}\right).

c

Find P\left(\dfrac{1}{2}\right).

10

Consider the polynomial P \left( x \right) = \left(x^{4} + 3\right) \left(2 - 4 x^{5}\right).

a

Find P \left( 0 \right).

b

Find P \left( -\dfrac{1}{3} \right).

c

Find P \left( 2 \right).

11

For the polynomial P(x)= 4 - \dfrac{7 x^{6}}{6}, find:

a

P(-1)

b

P(\sqrt{2})

c

P\left(\dfrac{1}{2}\right)

Equate coefficients
12

Let a x^{2} - 25 x + 25 = \left( 4 x - 5\right) \left(x - 5\right) for all values of x. Solve for a.

13

Find the values of A and B for each of the following equations:

a

Ax^2+Bx-8=5x^2-9x-8

b

A \left(x + 5\right) + B \left(x + 2\right) = 5 x + 16

c

x^{2} + 4 x + 8 = A \left(x + 2\right)^{2} + B

d

x^{3} + 6 x^{2} + 12 x + 10 = A \left(x + 2\right)^{3} + B

14

Find the values of A, \, B and C for each of the following equations:

a

x^{2} + 6 x + 13 = A \left(x + B\right)^{2} + C

b

5 x^{2} + 10 x + 9 = A \left(x + B\right)^{2} + C

c

3 x^{3} - 18 x^{2} + 36 x - 22 = A \left(x + B\right)^{3} + C

Operations on polynomials
15

Simplify the following:

a

\left( - 3 x^{2} - 8 x - 2\right) + \left( 5 x^{2} - 5 x + 5\right)

b

\left( - 8 x^{2} - 6\right) + \left( 3 x^{2} - 4 x\right)

c

\left( - 4 x^{3} - 4 x^{2} + 1\right) + \left( - x^{2} - 9 x - 5\right)

d

\left( 4 x^{2} y - 5 x y^{2} + 7\right) + \left( 9 x^{2} y + 9 x y^{2} - 4\right)

e

6 x^{3} - x^{2} + 5 x - \left(x^{3} + x + 2\right)

f

\left( - 8 x^{2} + 2 x - 8\right) - \left( 2 x^{2} - 6 x - 6\right)

g

\left( 7 x^{3} + 5 x^{2} + 8 x\right) - \left( 6 x^{3} + 7 x^{2} + 7 x + 9\right)

h

\left( - 6 x^{2} y - 4 x y^{2} + 2\right) - \left( - 3 x^{2} y + 5 x y^{2} - 9\right)

i

\left(x^{3} + 4 x^{2}\right) - \left( - 2 x + 9\right)

j

\left(x - 8\right)+\left( - 3 x^{3} + 6 x^{2} + 3 x-1\right)-\left( - 6 x^{2} + 5 x - 3\right)

k

3 \left( - 2 x + 7\right) - \left( 6 x^{2} + 8 x - 5\right) + 7 x

16

Let P \left(x\right) = x^{2} - 6 x + 6 and Q \left(x\right) = x + 7. Find the simplified expression of P \left(x\right) + Q \left(x\right).

17

Let P \left(x\right) = 2 x^{2} + 5 x - 7 and Q \left(x\right) = x + 2. Find the simplified expression of P \left(x\right) - Q \left(x\right).

18

Let A \left(x\right) = - 4 x^{2} + 6, B \left(x\right) = 6 x + 3 and C \left(x\right) = - 3 x^{2} - 5 x. Find the simplified expression of A \left(x\right) + B \left(x\right) + C \left(x\right).

19

Let A \left(x\right) = - 6 x^{2} + 5, B \left(x\right) = 3 x + 2 and C \left(x\right) = - x^{2} + x. Find the simplified expression of A \left(x\right) - B \left(x\right) - C \left(x\right).

20

Let P \left(x\right) = 2 x^{3} + 4 x^{2} - 8 x + 5, Q \left(x\right) = 9 - 8 x^{3} and R \left(x\right) = - 2 x^{2} + 9 x - x^{3} + 8. Find the simplified expression of the following:

a

P \left(x\right) - R \left(x\right)

b

P \left(x\right) - \left( R \left(x\right) - Q \left(x\right)\right)

21

Simplify the following:

a

\left( 3 b - 7\right)^{2}

b

\left(v + 5\right) \left( 5 v^{2} - 3 v - 5\right)

c

\left(m - 5\right) \left( 4 m - 3 + m^{2}\right).

d

\left( 5 u + 2\right) \left( - 2 u^{2} - 2 + 4 u\right)

e

\left( 5 c^{2} - 2\right) \left(3 - 2 c + 2 c^{2}\right)

f

\left( - 2 v - 4 + 2 v^{2}\right) \left( 5 v + 4\right)

g

\left( 4 y + 2\right) \left(y + 2\right)^{2}.

h

\left(x - 4\right) \left(x - 2\right) \left(x - 2\right)

i

\left( 3 u + 4\right) \left( - 2 u^{4} - 2 u^{2} + 3 u^{3} + 2 u\right)

j

\left( 4 u^{2} - 4 u + 5\right) \left( 5 u^{2} - 4 u - 1\right)

k

\left( 2 x^{3} + 3 x^{2} - 9 x + 6\right) \left(x - 3\right) - \left(x^{2} - 3 x - 1\right) \left(x - 3\right)

Applications
22

From 2006 to 2016, the population of Australia, measured in millions, could be estimated by the polynomial 0.02 x^{2} + 0.16 x + 20.6, where x is the number of years since 2006.

a

Estimate the population of Australia (in millions) in 2006, correct to one decimal place.

b

Use the polynomial to estimate the population of Australia (in millions) in 2009, correct to two decimal places.

c

According to the estimation given by the polynomial, how much (in millions) did the population of Australia increase by from 2006 to 2009? Round your answer to two decimal places.

23

Jenny's tool manufacturing business sells its tools exclusively through two retailers. The profit generated through selling at Bargains Bonanza is modelled by the equation \\A \left(v\right) = 2.6 v^{2} + 31.3 v + 880 and the profit generated through selling at Just Stuff is modelled by B \left(v\right) = 4.4 v^{2} - 40.3 v + 720, where v is the number of tools sold.

Form an expression for the polynomial, C \left(v\right), that models Jenny's total profit.

24

The revenue generated by Kathleen's cafe is modelled by R \left(y\right) = - 4.1 y^{2} + 31.6 y + 230 and the profit of her cafe is modelled by P \left(y\right) = 1.8 y^{2} + 40.4 y + 880, where y is the number of coffees produced.

Find the polynomial that models the costs of Kathleen's cafe.

25

If a picture frame has a length of (5 x^{2} - 4 x - 4)\text{ units} and a width of (6 x^{2} + 4 x)\text{ units}, form an expression for the perimeter of the rectangular picture frame.

26

Find the polynomial that represents the perimeter of the following shapes:

a
b
c
27

Find the polynomial that describes the total area of the given figure.

28

Consider a rectangular cardboard measuring 9\text{ cm} by 5\text{ cm}. It is to be converted into a box with no lid by cutting out square corners measuring x\text{ cm} in length and folding up the sides.

a

Find an expression for the length of the cardboard box in terms of x.

b

Find an expression for the width of the cardboard box in terms of x.

c

Form an expression in terms of x for the volume of the cardboard box. Give your answer in expanded form.

29

A landscape design company makes large raised boxes for plantings. The height of the boxes they build is (5 u + 3) \text{ m}, and the area of the rectangular base is (u^{2} + 2 u + 4)\text{ m}^2. If the box is filled with soil, what is the volume of soil needed?

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Outcomes

VCMNA357 (10a)

Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems.

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