Partial Fractions: Simple Linear Factors

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Michael Proudman

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From users who are not members of .Algebra :

joshua boddy

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Ready to use

8 years, 9 months ago

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joshua boddy 8 years, 9 months ago

Gave some feedback: Ready to use

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Michael Proudman 9 years, 5 months ago

Created this as a copy of Partial Fractions 1.

Name	Status	Author	Last Modified
Partial Fractions	draft	Bill Foster	06/08/2020 08:42
Partial Fractions 1	Ready to use	joshua boddy	01/06/2016 09:47
Partial Fractions 2	Ready to use	joshua boddy	01/06/2016 09:47
Partial Fractions 3 - double root	Ready to use	joshua boddy	01/06/2016 09:47
Integration with Partial Fractions 2	Ready to use	joshua boddy	01/06/2016 09:47
Integration by Partial Fractions 1 - double root	Has some problems	joshua boddy	23/02/2021 13:55
Partial Fractions: Simple Linear Factors	Ready to use	Michael Proudman	07/12/2021 10:59
Partial Fractions: Repeated Factor	Ready to use	Michael Proudman	07/12/2021 10:59
CF Maths In class test three mick paper question 7	Ready to use	Jinhua Mathias	28/08/2024 12:15
CF Maths January test mock paper Partial Fractions 3 - double root	Ready to use	Jinhua Mathias	28/08/2024 12:15
CF Maths Januaray test mock paper Partial Fractions 1	Ready to use	Jinhua Mathias	28/08/2024 12:15
Partial Fractions: Simple Linear Factors [L5]	draft	Abbi Mullins	25/07/2018 14:02
cormac's copy of CF Maths January test mock paper Partial Fractions 3 - double root	draft	cormac breen	30/10/2018 15:53
Blathnaid's copy of Partial Fractions: Simple Linear Factors	draft	Blathnaid Sheridan	13/11/2018 10:42
Blathnaid's copy of Michael's copy of Partial Fractions 3 - double root	draft	Blathnaid Sheridan	13/11/2018 10:43
Laura's copy of Partial Fractions: Repeated Factor	draft	Laura Horton	12/02/2019 16:22

There are 43 other versions that do you not have access to.

Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	9
c	integer	1
b	integer	-6
d	integer	-5
nb	string	adding
a1	integer	1
a2	integer	1
s1	integer	1
a_	integer	8
c_	integer	-3
b_	integer	-9
d_	integer	-7
nb_	string	taking away
a1_	integer	1
a2_	integer	1
s1_	integer	-1

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xxxxxxxxxx
 
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Generated value: `integer`

→ Used by:

Advice
"Part a)" - prompt
"Part a)" → "Unnamed gap" - Correct answer

Parts

Part a) Gap-fill
1. Gap Mathematical expression
  Add
Part b) 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.

Split

$\simplify{({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}$ into partial fractions.

Input the partial fractions here: Unnamed gap.

Advice

We use partial fractions to find $A$ and $B$ such that:

$\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \;\;\;=\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}$

Dividing both sides of the equation by $\displaystyle \simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\;\;$ we obtain:

$\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \Rightarrow \simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$

Identifying coefficients:

Constant term: $\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$

Coefficent $x$ : $\simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$

On solving these equations we obtain $A = \var{a}$ and $B=\var{c}$

Which gives:

$\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\;\;= \simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}$

We use partial fractions to find $A$ and $B$ such that:

$\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))} \;\;\;=\simplify[std]{ A/({a1_}x+{b_})+B/({a2_}x+{d_})}$

Dividing both sides of the equation by $\displaystyle \simplify[std]{1/( ({a1_}x+{b_})({a2_}x+{d_}) )}\;\;$ we obtain:

$\simplify[std]{A*({a2_}x+{d_})+B*({a1_}x+{b_}) = {a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}} \Rightarrow \simplify[std]{({a2_}A+{a1_}B)*x+{d_}*A+{b_}*B={a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}}$

Identifying coefficients:

Constant term: $\simplify[std]{ {d_}*A+{b_}*B={a_*d_+c_*b_} }$

Coefficent $x$ : $\simplify[std]{ {a2_}A+{a1_}B = {a_*a2_+c_*a1_} }$

On solving these equations we obtain $A = \var{a_}$ and $B=\var{c_}$

Which gives:

$\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))}\;\;= \simplify[std]{{a_}/({a1_}x+{b_})+{c_}/({a2_}x+{d_})}$

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joshua boddy 8 years, 9 months ago

Michael Proudman 9 years, 5 months ago

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Generated value: `integer`

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Partial Fractions: Simple Linear Factors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

joshua boddy 8 years, 9 months ago

Michael Proudman 9 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`