How do you write the partial fraction decomposition of the rational expression $\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}}$ $(9x^2 + 1)/(x^2(x − 2)^2)$ ?

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How do you write the partial fraction decomposition of the rational expression $\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}}$ $(9x^2 + 1)/(x^2(x − 2)^2)$ ?

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Answer 1 · 2018-08-11T07:24:27

$\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}} = \frac{1}{4 x} + \frac{1}{4 x^{2}} - \frac{1}{4 (x - 2)} + \frac{37}{4 {(x - 2)}^{2}}$ $(9x^2+1)/(x^2(x-2)^2) = 1/(4x)+1/(4x^2)-1/(4(x-2))+37/(4(x-2)^2)$

Explanation:

Given that we have squared linear factors in the denominator, the decomposition will take a form like:

$\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 2} + \frac{D}{{(x - 2)}^{2}}$ $(9x^2+1)/(x^2(x-2)^2) = A/x+B/x^2+C/(x-2)+D/(x-2)^2$

Multiplying both sides by $x^{2} {(x - 2)}^{2}$ $x^2(x-2)^2$ this becomes:

$9 x^{2} + 1 = A x {(x - 2)}^{2} + B {(x - 2)}^{2} + C x^{2} (x - 2) + D x^{2}$ $9x^2+1 = Ax(x-2)^2+B(x-2)^2+Cx^2(x-2)+Dx^2$

Putting $x = 0$ $x=0$ we get:

$1 = 4 B$ $1 = 4B" "$ so $B = \frac{1}{4}$ $" "B = 1/4$

Putting $x = 2$ $x=2$ we get:

$37 = 4 D$ $37 = 4D" "$ so $D = \frac{37}{4}$ $" "D=37/4$

Looking at the coefficient of $x$ $x$ , we have:

$0 = 4 A - 4 B = 4 A - 1$ $0 = 4A-4B = 4A-1" "$ and hence $A = \frac{1}{4}$ $A=1/4$

Looking at the coefficient of $x^{3}$ $x^3$ , we have:

$0 = A + C$ $0 = A+C" "$ and hence $C = - A = - \frac{1}{4}$ $C=-A=-1/4$

So:

$\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}} = \frac{1}{4 x} + \frac{1}{4 x^{2}} - \frac{1}{4 (x - 2)} + \frac{37}{4 {(x - 2)}^{2}}$ $(9x^2+1)/(x^2(x-2)^2) = 1/(4x)+1/(4x^2)-1/(4(x-2))+37/(4(x-2)^2)$

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How do you write the partial fraction decomposition of the rational expression $\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}}$ $(9x^2 + 1)/(x^2(x − 2)^2)$ ?

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Explanation:

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How do you write the partial fraction decomposition of the rational expression (9x^2 + 1)/(x^2(x − 2)^2)9x2+1x2(x−2)2 (9x^2 + 1)/(x^2(x − 2)^2)?

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Explanation:

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How do you write the partial fraction decomposition of the rational expression $\frac{9 x^{2} + 1}{x^{2} {(x - 2)}^{2}}$ $(9x^2 + 1)/(x^2(x − 2)^2)$ ?