AE6 Partial Fractions

Error

There was an error loading the page.

Contributors

Feedback

From users who are members of Skills Audits for Maths and Stats :

Will Morgan	said	Ready to use	1 year, 2 months ago
Oliver Spenceley	said	Needs to be tested	1 year, 3 months ago

History

○

Will Morgan 1 year, 1 month ago

Published this.

○

Will Morgan 1 year, 2 months ago

Gave some feedback: Ready to use

○

Oliver Spenceley 1 year, 3 months ago

Gave some feedback: Needs to be tested

○

Oliver Spenceley 1 year, 3 months ago

Created this as a copy of Partial Fractions: Proper Fractions 7c.

Name	Status	Author	Last Modified
Partial Fractions: Proper Fractions 5a	Ready to use	Ben McGovern	08/08/2023 14:27
Partial Fractions: Proper Fractions 5b	Ready to use	Ben McGovern	08/08/2023 14:43
Partial Fractions: Proper Fractions 5c	Ready to use	Ben McGovern	14/08/2023 12:00
Partial Fractions: Proper Fractions 6a	Ready to use	Ben McGovern	10/08/2023 14:21
Partial Fractions: Proper Fractions 7b	Ready to use	Ben McGovern	10/08/2023 15:50
Partial Fractions: Proper Fractions 6c	Ready to use	Ben McGovern	14/08/2023 12:26
Partial Fractions: Proper Fractions 7a	Ready to use	Ben McGovern	10/08/2023 15:45
Partial Fractions: Proper Fractions 6b	Ready to use	Ben McGovern	10/08/2023 14:29
Partial Fractions: Proper Fractions 7c	Ready to use	Ben McGovern	14/08/2023 12:38
Integration with Partial Fractions 5	Ready to use	Ben McGovern	27/09/2022 09:01
Integration with Partial Fractions 6	Ready to use	Ben McGovern	27/09/2022 09:01
Integration with Partial Fractions 7	Ready to use	Ben McGovern	27/09/2022 09:01
Integration with Partial Fractions 8	Ready to use	Ben McGovern	27/09/2022 09:01
Integration with Partial Fractions 9	Ready to use	Ben McGovern	27/09/2022 09:01
Integration with Partial Fractions 10	Ready to use	Ben McGovern	27/09/2022 09:01
AE6 Partial Fractions	Ready to use	Oliver Spenceley	09/04/2024 15:18

There are 36 other versions that do you not have access to.

Question statement

Rewrite the following expression as partial fractions:

$\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))}.$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-3
pairs	list	Nested 5×2 list
index	integer	0
b	integer	-4
c	integer	1
m	integer	1
k	integer	3
n	integer	-5
Asol	number	1.5
Bsol	number	-0.5
Csol	number	-0.5
check	string	<p>\[ \simplify{({m}x^2+{n}x+{
sol1	string	<p>\[ \simplify{({m}x^2+{n}x+{
sol2	string	<p>\[ \simplify{({m}x^2+{n}x+{
sol3	string	<p>\[ \simplify{({m}x^2+{n}x+{
simp1	number	1
simp2	number	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-6..6 except 0)

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`

-3

→ Used by:

Asol
Bsol
Csol
simp1
simp2
sol1
sol2
sol3

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

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

Unnamed gap

Advice

To express

$\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))}$ as partial fractions, we want to set this equal to the sum of two fractions with denominators

$\simplify{x+{a}}$ and

$\simplify{x^2+{b}x+{c}}$ . Since we have a linear factor and a quadratic factor, this tells us that the form of the partial fractions will be

$\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \simplify{A/(x+{a}) + (B*x+C)/(x^2+{b}x+{c})},$

where $A$ , $B$ , and $C$ are constants.

To find the values of $A$ , $B$ , and $C$ , we want to first multiply this equation by the denominator of the left-hand side. This gives

$\simplify{{m}x^2+{n}x+{k}=A(x^2+{b}x+{c})+B*x(x+{a}) + C(x+{a})}.$

(Note: To find $A$ , $B$ , and $C$ , we will use a combination of choosing suitable values of $x$ to eliminate terms, and equating coefficients. It can be solved by only equating coefficients, but this is a more efficient process.)

To find $A$ , we can eliminate $B$ and $C$ by setting $x=\var{-a}$ :

$\simplify{{m*a^2-n*a+k}=A{(a^2-b*a+c)}} \implies A=\simplify[fractionNumbers]{{Asol}}.$

To find $C$ , we can eliminate $B$ by setting $x=0$ and substituting in the result of $A$ :

$\simplify{{k}={c}A+{a}C} \implies C=\simplify[all,fractionNumbers]{({k}-{c}A)/{a}}.$

Hence,

$C = \simplify[fractionNumbers]{{Csol}}.$

Finally, by equating coefficients of the $x^2$ -terms we can find $B$ :

$(x^2): \quad \var{m} = \simplify{A+B} \implies B=\var{m}-A.$

Therefore,

$B=\simplify[fractionNumbers]{{Bsol}},$

and

{check}

Use this link to find some resources which will help you revise this topic.

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
Mathematical expression
Mathematical expression		Hasn't run yet

This question is used in the following exams:

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 1 year, 1 month ago

Will Morgan 1 year, 2 months ago

Oliver Spenceley 1 year, 3 months ago

Oliver Spenceley 1 year, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

AE6 Partial Fractions

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Will Morgan 1 year, 1 month ago

Will Morgan 1 year, 2 months ago

Oliver Spenceley 1 year, 3 months ago

Oliver Spenceley 1 year, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`