005 Sample Final A, Question 10

Question Write the partial fraction decomposition of the following,

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{x+2}{x^3-2x^2+x}}

Step 1:
First, we factor the denominator. We have $x^{3}-2x^{2}+x=x(x^{2}-2x+1)=x(x-1)^{2}$

Step 2:
Since we have a repeated factor in the denominator, we set ${\frac {x+2}{x(x-1)^{2}}}={\frac {A}{x}}+{\frac {B}{x-1}}+{\frac {C}{(x-1)^{2}}}$.

Step 3:
Multiplying both sides of the equation by the denominator $x(x-1)^{2}$, we get
$x+2=A(x-1)^{2}+B(x)(x-1)+Cx$.

Step 4:
If we let $x=0$, we get $2=A$. If we let $x=1$, we get $3=C$.

Step 5:
To solve for $B$, we plug in $A=2$ and $C=3$ and simplify. We have
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x+2=2(x-1)^{2}+B(x)(x-1)+3x=2x^{2}-4x+2+Bx^{2}-Bx+3x} . So, $x+2=(2+B)x^{2}+(-1-B)x+2$. Since both sides are equal,
we must have 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 2+B=0} and 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 -1-B=1} . So, 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 B=2} . Thus, the decomposition is 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{x+2}{x(x-1)^2}=\frac{2}{x}+\frac{2}{x-1}+\frac{3}{(x-1)^2}} .

Final Answer:
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{x+2}{x(x-1)^2}=\frac{2}{x}+\frac{2}{x-1}+\frac{3}{(x-1)^2}}