Partial Fractions: Proper Fractions 5c

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Created this as a copy of Partial Fractions: Proper Fractions 5b.

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

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Question statement

Rewrite the following expression as partial fractions:

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-6
pairs	list	Nested 5×2 list
index	integer	2
b	integer	1
c	integer	3
n	integer	4
Asol	number	0.0888888889
Bsol	number	-0.0888888889
Csol	number	-0.6222222222
check	string	<p>\[ \simplify{{n}/((x+{a})(x
sol1	string	<p>\[ \simplify{{n}/((x+{a})(x
sol2	string	<p>\[ \simplify{{n}/((x+{a})(x
sol3	string	<p>\[ \simplify{{n}/((x+{a})(x
simp1	number	4
simp2	number	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-6..6 except 0)

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Description

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Generated value: `integer`

-6

This variable doesn't seem to be used anywhere.

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1. Gap Mathematical expression
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Advice

To express

$\simplify{{n}/((x+{a})(x^2+{b}x+{c}))}$ in terms of 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

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

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

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

(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 $x=\var{-a}$ :

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

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

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

Hence,

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

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

$(x^2): \quad 0 = \simplify{A+B} \implies B=-A.$

Therefore,

$B=\simplify[fractionNumbers]{-{Asol}},$

and

{check}

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This question is used in the following exams:

Partial Fractions by Ben McGovern in MASH Bath: Moodle quizzes and TS.
All the good stuff by Chris Templet in Chris's workspace.
PF by Dean Ang in Dean's workspace.

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Ben McGovern 1 year, 8 months ago

Lucia Sagredo 2 years, 5 months ago

Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ben McGovern 4 years, 6 months ago

Error in variable testing condition

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Generated value: `integer`

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Partial Fractions: Proper Fractions 5c

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben McGovern 1 year, 8 months ago

Lucia Sagredo 2 years, 5 months ago

Ben McGovern 2 years, 7 months ago

Ben McGovern 3 years, 9 months ago

Ben McGovern 4 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

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Generated value: `integer`