Partial Fractions - quadratic term in denominator

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Name	Status	Author	Last Modified
Partial Fractions - quadratic term in denominator	Ready to use	joshua boddy	01/06/2016 09:47
Partial Fractions: Quadratic Term in Denominator	Ready to use	Michael Proudman	07/12/2021 10:59

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Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

coeff0	integer	6
coeff1	integer	2
a	integer	1
a1	integer	5
a2	integer	3
a3	integer	-4
bcoeff0	integer	7
bcoeff1	integer	1
b	integer	-2
b1	integer	3
b2	integer	-3
b3	integer	-3
ccoeff0	integer	5
ccoeff1	integer	1
c	integer	2
c1	integer	4
c2	integer	4
c3	integer	-4

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1. Gap Mathematical expression
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Part b) Gap-fill
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Part c) Gap-fill
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Express the following as a sum of partial fractions

$\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$

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Suppose we want to express

$\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}$

as the sum of its partial fractions.

When the denominator contains a quadratic factor we have to consider the possibilty that the numebrator can contain a term in $x$ . This is because if it did, the numerator would still be of lower degree than the denominator - this would be a proper fraction. So we write

$\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})/((x^2+{coeff1}x+{coeff0})(x-{a}))}=\simplify{(Ax+B)/(x^2+{coeff1}x+{coeff0})+c/(x+{a})}$

We multiply both sides by $\simplify{((x^2+{coeff1}x+{coeff0})(x-{a}))}$ to give $\simplify{({a1+a3}x^2+{a2-a*a1+a3*coeff1}x+{a3*coeff0-a2*a})}=\simplify{(Ax+B)(x+{a})+C(x^2+{coeff1}x+{coeff0})}$

By evaluating both sides at $\simplify{x=-{a}}$ we can find C $=\var{a3}$

Then by comparing coefficients we find A $=\var{a1}$ and B $=\var{a2}$

Apply the same method for b) and c)

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This question is used in the following exams:

Partial Fractions by Deirdre Casey in Deirdre's workspace.
Partial Fractions Tutorial by Julie Crowley in Julie's workspace.

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Partial Fractions - quadratic term in denominator

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joshua boddy 9 years, 6 months ago

joshua boddy 9 years, 6 months ago

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Generated value: `integer`

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Partial Fractions - quadratic term in denominator

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

joshua boddy 9 years, 6 months ago

joshua boddy 9 years, 6 months ago

Error in variable testing condition

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Generated value: integer

Parts

Gap-fill

Generated value: `integer`