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1.10 Solve cubic equations

Worksheet
Solving cubic equations
1

Solve the following cubic equations:

a

x^{3} = - 8

b

x^{3} - 49 x = 0

c

x^{3} - 125 = 0

d

- 3 x^{3} = 5 x^{2}

e

8 x^{3} - 125 = 0

f

\left(x + 8\right) \left(x + 4\right) \left(1 + x\right) = 0

g

\left( 5 x - 4\right) \left(x + 3\right) \left(x - 2\right) = 0

h

512 x^{3} - 125 = 0

i

729 x^{3} + 8 = 0

j

x^{3} + x^{2} - 20x = 0

k

x^{3} - 5 x^{2} - 4 x + 20 = 0

l

x \left(x + 7\right) \left(x + 8\right) = 9 \left(x + 7\right) \left(x + 8\right)

m

x^{3} - 4 x^{2} - 45 x = 0

n

x^{3} + 9 x^{2} + 27 x + 27 = 0

o

- 64 x^{3} + 48 x^{2} - 12 x + 1

p

x^{3} - 5 x^{2} - 49 x + 245 = 0

q

x^{3} + 13 x^{2} + 47 x + 35 = 0

r

150 x^{3} + 115 x^{2} - 118 x - 56 = 0

s

x^{3} - 3 x^{2} - 18 x + 40 = 0

t

x^{3} - 4 x^{2} - 45 x = 0

u

x^{3} + 9 x^{2} + 27 x + 27 = 0

v

- 64 x^{3} + 48 x^{2} - 12 x + 1

w

x^{3} - 5 x^{2} - 49 x + 245 = 0

x

x^{3} + 13 x^{2} + 47 x + 35 = 0

y

150 x^{3} + 115 x^{2} - 118 x - 56 = 0

z

x^{3} - 3 x^{2} - 18 x + 40 = 0

2

Consider the equation x^{3} - 512 = 0.

a

Find a value of x that satisfies x^{3} = 512.

b

Find a factorisation of x^{3} - 512 as a product of a linear and a quadratic factor.

c

How many zeros does the quadratic factor have?

d

Hence, how many solutions does x^{3} - 512 = 0 have?

3

The cubic P \left( x \right) = x^{3} - 7 x^{2} + 14 x - 8 has a factor of x - 1. Solve for the roots of the cubic.

4

A cubic function is defined as y = x \left(x - 2\right) \left(x - 1\right). Solve for the roots of the cubic.

5

What is the double root of the function y = 10 x^{2} - x^{3} ?

6

One of the solutions of the equation \left( a x + 4\right) \left(x + 6\right) \left(x - 7\right) = 0 is x = - 4.

Solve for the value of a.

7

The expression 3 x^{3} - 4 x^{2} - 5 x + 2 = 0 has a factor of x + 1.

a

Fully factorise 3 x^{3} - 4 x^{2} - 5 x + 2.

b

Hence, solve the equation 3 x^{3} - 4 x^{2} - 5 x + 2 = 0.

8

The polynomial x^{3} - 2 x^{2} - 5 x + 6 has a factor of x - 3.

a

Find the quadratic factor that is multiplied by x - 3 to get x^{3} - 2 x^{2} - 5 x + 6.

b

Hence, solve the equation x^{3} - 2 x^{2} - 5 x + 6 = 0.

9

Consider the equation x^{3} + 4 x^{2} - 39 x - 126 = 0.

a

Given that one solution of the equation is x = - 7, identify one linear factor of

x^{3} + 4 x^{2} - 39 x - 126.

b

Find the quadratic factor that is multiplied by x + 7 to get x^{3} + 4 x^{2} - 39 x - 126.

c

Hence, solve the equation x^{3} + 4 x^{2} - 39 x - 126 = 0.

Applications
10

The volume of a sphere is given by the formula V = \dfrac{4}{3} \pi r^{3}.

If a sphere of radius r\text{ m} has volume 288 \pi\text{ m}^3, find the value of r.

11

The mass (M) in kilograms of a cubic container of water is given by M = 0.001 r^{3}, where r is the side length of the cube-shaped container in centimetres.

The greatest mass Ivan can carry is equivalent to his own weight, which is 73 kilograms. What is the length (in \text{cm}) of the largest cubic container of water Ivan can carry correct to two decimal places?

12

The volume of a reservoir, in \text{mL}, during the first half of the year can be described by the following equation, where 0 \leq t \leq 6 is time in months:

V \left( t \right) = - t^{3} + 30 t^{2} - 131 t + 350

During which months did the reservoir contain 200 \text{ mL}? (Note for January: t=1)

13

The population of bacteria in a slow-growing culture t hours after 7 am can be described by the equation P \left( t \right) = 200 + 15 t + t^{3}.

a

What is the initial number of bacteria at 7 am?

b

How many hours does it take for the initial number of bacteria to double?

c

Once the number of bacteria reaches 1350, the experiment is stopped. At what time of the day does this happen?

14

A retail outlet sells x units of a product each year. The annual cost for this product is given by C = 9 x + 3 x^{2} and the annual gross profit is given by G = 9 - 2 x - 4 x^{2} + x^{3}, where C and G are both in dollars.

a

Form an expression for P, the net profit after producing and selling x units.

b

How many units must be sold to make a profit of \$40\,672?

15

A cylindrical can is to be designed using a fixed amount of tin material, so that the total surface area including the top and bottom faces is 150 \pi \text{ cm}^2 .

a

If the radius of the cylinder is represented by r, and the height of the cylinder is represented by h, express h in terms of r.

b

If the volume of the can is represented by V, form an expression for V in terms of r only.

c

What is the radius of the cylinder if the volume is 250 \pi\text{ cm}^3 ?

16

A box is formed by cutting squares of length x\text{ cm} from the corners of a piece of cardboard 10\text{ cm} by 30\text{ cm}.

a

Find an expression for the volume of the box in terms of x, in expanded form.

b

What is the value of x if the volume is 176 \text{ cm}^3?

17

To solve a cubic equation of the form x^{3} + m x = n, we can use the formula: x = \sqrt[3]{\dfrac{n}{2} + \sqrt{\left(\dfrac{n}{2}\right)^{2} + \left(\dfrac{m}{3}\right)^{3}}} - \sqrt[3]{\dfrac{- n}{2} + \sqrt{\left(\dfrac{n}{2}\right)^{2} + \left(\dfrac{m}{3}\right)^{3}}}

Use this formula to find the real solution of x^{3} + 24 x = 56.

Using technology
18

Use technology to solve the following cubic equations:

a

147 x^{3} + 427 x^{2} + 160 x - 84 = 0

b

50 x^{3} + 155 x^{2} + 152 x + 48 = 0

c

64 x^{3} + 104 x^{2} + 78 x + 18 = 0

d

48 x^{3} - 212 x^{2} + 84 x + 281 = - 36

e

- 576 x^{3} + 272 x^{2} - 41 x - 74 = 76

f

- 25 x^{3} + 20 x^{2} - 5 x + 85 = - 75

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Outcomes

ACMMM019

solve cubic equations using technology, and algebraically in cases where a linear factor is easily obtained

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