# 4.09 Polynomial long division

Worksheet
Polynomial division
1

Simplify the following:

a

\dfrac{\left( n^{7} r^{2}\right)^{4}}{\left( n^{4} r\right)^{4}}

b

\dfrac{9 x^{5} + 8 x^{4}}{x}

c

\dfrac{40 x^{3} - 48 x^{2} + 32}{8}

d

\dfrac{15 x^{4} - 24 x^{3} - 21 x^{2} + 12 x}{3 x}

2

Complete the following statement:

\dfrac{⬚ x^{7} - ⬚ x^{5}}{2 x^{⬚}} = 3 x^{4} - 4 x^{2}

3

What polynomial, when divided by 2 x^{2}, produces 8 x^{6} - 5 x^{4} + 6 x^{2} as a quotient?

4

Divide x^2 - 6x + 1 by x + 3 using long division.

a

State the remainder of this division.

b

State the quotient of this division.

5

Perform the following using long division. State the quotient and remainder.

a
x^2 + 3x + 2 \div x - 3
b

\left(x^{2} - 5 x + 11\right) \div \left(x + 5\right)

c

\left( 3 x^{3} - 15 x^{2} + 2 x - 10\right) \div \left(x - 5\right)

d

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

6

Consider the division: \dfrac{x^{2} + 17 x + 70}{x + 10}.

a

Use polynomial division to determine whether x + 10 a factor of the polynomial

x^{2} + 17 x + 70.

b

Find the other factor.

Remainder theorem
7

For each of the following, find the remainder when P \left( x \right) is divided by A \left( x \right):

a

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

b

P \left( x \right) = 3 x^{4} + 5 x^{3} - 2 x^{2} + 6 x + 7, A \left( x \right) = 3 x + 1

c

P \left( x \right) = 2x^4-3x^3+6x^2-10, A \left( x \right) = x -1

d

P \left( x \right) = x^3-2x^2+9x-1, A \left( x \right) = x-4

e

P \left( x \right) = 4x^5-6x^3-7x^2+9x, A \left( x \right) = 2x-5

f

P \left( x \right) = 6x^4-x^3+9x^2+10x-8, A \left( x \right) = 4x+2

8

Find the value of k for each of the following:

a

The remainder when 3 x^{3} + 4 x^{2} + 4 x + k is divided by x - 2 is 52.

b

The remainder when 4 x^{3} - 2 x^{2} + k x-1 is divided by x - 2 is 15.

9

Write down all the possible rational zeros of the following polynomials:

a

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

b

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

c

P \left( x \right) = x^{3} + 4 x^{2} - 7 x - 10

10

Is \dfrac{2}{5} a possible rational zero of P \left( x \right) = 5 x^{3} - 4 x^{2} - 7 x + 10?

11

The polynomials P \left( x \right) = x^{3} + 2 x^{2} - 5 x + n and Q \left( x \right) = x^{3} + 4 x - 11 give the same remainder when divided by x - 4. Solve for n.

12

Consider the polynomials P \left( x \right) = x^{4} - 5 x^{3} - k x + m and Q \left( x \right) = k x^{2} + m x - 5. The remainder when P \left( x \right) is divided by x + 2 is 53, while the remainder when Q \left( x \right) is divided by x + 2 is - 31.

a

Solve for k.

b

Solve for m.

c

Hence, find the remainder when P \left( x \right) is divided by x - 4.

Factor theorem
13

Write the following polynomials as a product of linear factors:

a

x^{3} - 6 x^{2} + 11 x - 6

b

4 x^{3} - x^{2} - 29 x + 30

14

Consider the polynomial P \left( x \right) = x^{3} - 4 x^{2} - 11 x + 30.

a

Write down all the possible zeros.

b

Find the value of P \left( - 1 \right).

c

Find the value of P \left( - 3 \right).

d

Find the value of P \left( 2 \right).

e

Factorise P \left( x \right) = x^{3} - 4 x^{2} - 11 x + 30.

15

Consider the division \dfrac{4 x^{2} - 3 x - 6}{x - 2}.

a

Find the remainder.

b

Is x - 2 a factor of P \left( x \right)?

16

Consider the division \dfrac{x^{3} - 5 x^{2} - 2 x - 1}{x + 1}.

a

Find the remainder.

b

Is x + 1 a factor of P \left( x \right) ?

17

Consider the division \dfrac{x^{2} + 4 x - 32}{x + 8}.

a

Find the remainder.

b

Is x + 8 a factor of P \left( x \right)?

c

Factorise x^{2} + 4 x - 32.

18

Show that x + 2 is a factor of P \left( x \right) = x^{4} + 7 x^{3} + 8 x^{2} - 28 x - 48.

19

Consider \left( 4 x^{3} + 20 x^{2} + 3 x + 15\right) \div \left(x + 5\right).

a

Show that x + 5 is a factor of P \left( x \right).

b

Factorise 4 x^{3} + 20 x^{2} + 3 x + 15.

20

Consider \left(12 + 9 x + x^{2} - x^{3}\right) \div \left(4 - x\right).

a

Show that 4 - x is a factor of P \left( x \right).

b

Factorise 12 + 9 x + x^{2} - x^{3}.

21

The polynomial P \left( x \right) = x^{3} + a x^{2} + b - 10 x is divisible by both x+1 and x+2.

a

Solve for the value of a.

b

Hence, solve for the value of b.

22

The polynomials 4 x^{2} - 13 x - 12 and 5 x^{2} + 11 x + k have a common factor of x + p, where p is an integer.

a

Solve for p.

b

Hence, solve for k.

23

The polynomial 3 x^{3} + p x^{2} + q x + 2 has a factor of x + 1, but when divided by x - 1, it leaves a remainder of 24.

a

Solve for p.

b

Solve for q.

c

Hence, factorise the polynomial completely.

24

The polynomial P \left( x \right) = x^{3} + a x^{2} + b + 40 x is divisible by both x+3 and x+4.

a

Solve for a.

b

Hence, solve for b.

25

The polynomial 3 x^{3} + p x^{2} + 8 x + q is divisible by x^{2} - 7 x + 12.

a

Solve for p.

b

Solve for q.

c

Hence, factorise the cubic completely.

26

The polynomial Q \left( x \right) = x^{4} + 8 x^{3} + a x^{2} - 74 x + b is divisible by both x - 3 and x + 5.

a

Solve for a.

b

Solve for b.

c

Hence, factorise Q \left( x \right) completely.

Applications
27

Consider the rectangle shown with an area of (2 x^{4} - 8 x)\text{ units}^2:

Find a polynomial expression for its length.

28

Consider the rectangle shown with an area of (21 x^{3} + 6 x^{2} - 15 x - 9)\text{ units}^2:

Find a polynomial expression for its length.

29

The rectangle shown has an area of \\ 5 x^{3} + 7 x^{2} - 18 x - 8\text{ units}^2:

Find a polynomial expression for its length.

30

For the space station, an engineer has designed a new rectangular solar panel that has an area of \left( 12 x^{3} - 28 x^{2} + 21 x - 5\right)\text{ m}^2. The width of the solar panel is \left( 2 x^{2} - 3 x + 1\right)\text{ m}.

Find an expression for the length of the solar panel.

31

Consider the triangle shown with the given area of (19 n^{3} + 13 n^{2} + 11 n)\text{ units}^2.

Find a polynomial expression for its height.

32

This parallelogram has an area of \\ 2 x^{3} + 4 x^{2} - 5 x - 1 \text{ units}^2:

Find a polynomial expression for the length of its base.

33

If the distance travelled is \left( 3 x^{3} - 2 x^{2} + 4 x + 9\right)\text{ km} and the speed is \left(x + 1\right)\text{ km/h}, find the time travelled in hours.

34

It costs \left( 4 x^{5} + 5 x^{4} + 2 x^{3} + 11 x^{2} - 3 x + 14\right) dollars to replace the lawn in the backyard. If the new lawn costs \left(x + 2\right) dollars per square metre, what is the area of the lawn in square metres?