Cubics: Solving a Cubic Equation

Error

There was an error loading the page.

Contributors

Feedback

From users who are members of MASH Bath: Question Bank :

Ben McGovern

said

Ready to use

3 years, 9 months ago

History

○

Ben McGovern 2 years, 7 months ago

Published this.

○

Ben McGovern 3 years, 9 months ago

Gave some feedback: Ready to use

○

Ben McGovern 4 years, 9 months ago

Created this as a copy of Cubics: Calculating Coefficients.

Name	Status	Author	Last Modified
Cubics: Calculating Coefficients	Ready to use	Ben McGovern	14/09/2022 15:44
Cubics: Solving a Cubic Equation	Ready to use	Ben McGovern	14/08/2023 15:43

There are 5 other versions that do you not have access to.

Question statement

Solve the cubic equation

$\simplify{x^3+{a+b+c}x^2+{a*b+a*c+b*c}x+{a*b*c}=0}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-1
b	integer	-3
c	integer	-8

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
-random(1,2)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-1

→ Used by:

Statement
Advice
"Unnamed part" → "Unnamed gap" - Correct answer

Parts

Gap-fill
1. Gap List of numbers
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

$x=$ Unnamed gap (Separate your answers with a comma.)

Advice

To solve

$\simplify{x^3+{a+b+c}x^2+{a*b+a*c+b*c}x+{a*b*c}=0},$

we want to factorise the left-hand side of the equation into its linear factors.

To find the first of these factors, we can try appropriate values for $x$ to see if they satisfy the above equation. A sensible approach to this step is to try the positive and negative factors of the constant term in the equation, which is $\var{a*b*c}$ in this case.

Starting with $\var{-a}$ , we find this does satisfy the equation, which tells us that $(\simplify{x+{a}})$ is a factor of the polynomial.

(Note: The first number you try will often not be a factor, and this is completely fine. It can be helpful to list the factors first and try each one in order until you find one that works.)

To find the other factors, we can write the left-hand side as a product of this factor and a quadratic, and then factorise the quadratic to find the remaining linear factors:

$\begin{split} \simplify[all]{x^3+{a+b+c}x^2+{a*b+a*c+b*c}x+{a*b*c}} &\,= \simplify{(x+{a})(x^2+{b+c}x+{b*c})} \\ &\,= \simplify{(x+{a})(x+{b})(x+{c})}. \end{split}$

Therefore,

$\simplify{(x+{a})(x+{b})(x+{c}) = 0},$

and the solutions for $x$ are $x_1 = \var{-a}$ , $x_2 = \var{-b}$ and $x_3 = \var{-c}$ .

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
List of numbers
List of numbers		Hasn't run yet

This question is used in the following exams:

Cubic Equations by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Cubics: Solving a Cubic Equation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ben McGovern 4 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`