Difference between revisions of "004 Sample Final A, Problem 10"

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! Foundations
 
! Foundations
 
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|1) What is the form of the partial fraction decomposition of <math>\frac{3x-37}{(x+1)(x-4)}</math>?
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|2) What is the form of the partial fraction decomposition of <math>\frac{4x^2}{(x-1){(x-2)}^2}</math>?
 
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|Answer:
 
|Answer:
 
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|1) <math>\frac{A}{x+1}+\frac{B}{x-4}</math>
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|2)<math>\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}</math>
 
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! Step 1:
 
! Step 1:
 
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|We set <math>\frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}=\frac{A}{x-1}+\frac{B}{x+3}+\frac{C}{{(x+3)}^2}</math>.
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! Step 2:
 
! Step 2:
 
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|Multiplying both sides of the equation by <math>(x + 3)^2(x-1)</math>, we get
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|<math>6x^2+27x+31=A(x+3)^2+B(x+3)(x-1)+C(x-1)</math>.
 
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Revision as of 16:33, 4 May 2015

Decompose into separate partial fractions.     

Foundations
1) What is the form of the partial fraction decomposition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{3x-37}{(x+1)(x-4)}} ?
2) What is the form of the partial fraction decomposition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4x^2}{(x-1){(x-2)}^2}} ?
Answer:
1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{x+1}+\frac{B}{x-4}}
2)Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}}


Solution:

Step 1:
We set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}=\frac{A}{x-1}+\frac{B}{x+3}+\frac{C}{{(x+3)}^2}} .
Step 2:
Multiplying both sides of the equation by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x + 3)^2(x-1)} , we get
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6x^2+27x+31=A(x+3)^2+B(x+3)(x-1)+C(x-1)} .
Step 3:
Step 4:
Final Answer:

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