MathAdmin at 17:20, 2 June 2015

2015-06-02T17:20:45Z

MathAdmin: Created page with " Decompose into separate partial fractions. $\frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}$ {| class="mw-collapsible mw-collaps..."

2015-06-01T05:48:43Z

Created page with "<span class="exam"> Decompose into separate partial fractions. <math>\frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}</math> {| class="mw-collapsible mw-collaps..."

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<span class="exam"> Decompose into separate partial fractions.      <math>\frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}</math>
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Foundations
|-
|1) What is the form of the partial fraction decomposition of <math>\frac{3x-37}{(x+1)(x-4)}</math>?
|-
|2) What is the form of the partial fraction decomposition of <math>\frac{4x^2}{(x-1){(x-2)}^2}</math>?
|-
|Answer:
|-
|1) <math>\frac{A}{x+1}+\frac{B}{x-4}</math>
|-
|2)<math>\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}</math>
|}

Solution:

{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
|-
|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>.
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 2:
|-
|Multiplying both sides of the equation by <math>(x + 3)^2(x-1)</math>, we get
|-
|<math>6x^2+27x+31=A(x+3)^2+B(x+3)(x-1)+C(x-1)</math>.
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
|-
|If we set <math>x=1</math> in the above equation, we get <math>16A=64</math> and <math>A=4</math>.
|-
|If we set <math>x=-3</math> in the above equation, we get <math>-4C=4</math> and <math>C=-1</math>.
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 4:
|-
|In the equation <math>6x^2+27x+31=A(x+3)^2+B(x+3)(x-1)+C(x-1)</math>, we compare the constant terms of both sides. We must have
|-
|<math>9A-3B-C=31</math>. Substituting <math>A=4</math> and <math>C=-1</math>, we get <math>B=2</math>.
|-
|Thus, the partial fraction decomposition is <math>\frac{4}{x-1}+\frac{2}{x+3}+\frac{-1}{{(x+3)}^2}</math>
|}

{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
|-
|<math>\frac{4}{x-1}+\frac{2}{x+3}+\frac{-1}{{(x+3)}^2}</math>
|}

[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]

@@ Line 11: / Line 11: @@
 |1) <math>\frac{A}{x+1}+\frac{B}{x-4}</math>
 |-
-|2)<math>\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}</math>
+|2) <math>\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{{(x-2)}^2}</math>
 |}

004 Sample Final A, Problem 10 - Revision history

MathAdmin at 17:20, 2 June 2015

MathAdmin: Created page with " Decompose into separate partial fractions. \frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)} {| class="mw-collapsible mw-collaps..."

MathAdmin: Created page with " Decompose into separate partial fractions. $\frac{6x^2 + 27x + 31}{(x + 3)^2(x-1)}$ {| class="mw-collapsible mw-collaps..."