How do you write the partial fraction decomposition of the rational expression $\frac{x^{4} + 6}{x^{5} + 7 x^{3}}$ $(x^4+6)/(x^5+7x^3)$ ?

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{4} + 6}{x^{5} + 7 x^{3}}$ $(x^4+6)/(x^5+7x^3)$ ?

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Answer 1 · 2016-11-14T07:35:55

The answer is $= \frac{- \frac{6}{49}}{x} + \frac{\frac{6}{7}}{x^{3}} + \frac{\frac{55}{49} x}{x^{2} + 7}$ $=(-6/49)/x+(6/7)/x^3+(55/49x)/(x^2+7)$

Explanation:

$\frac{x^{4} + 6}{x^{5} + 7 x^{3}} = \frac{x^{4} + 6}{(x^{3}) (x^{2} + 7)}$ $(x^4+6)/(x^5+7x^3)=(x^4+6)/((x^3)(x^2+7))$

Let's do the decompositiom in partial fractions

$\frac{x^{4} + 6}{(x^{3}) (x^{2} + 7)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x^{3}} + \frac{D x + E}{x^{2} + 7}$ $(x^4+6)/((x^3)(x^2+7))=A/x+B/x^2+C/x^3+(Dx+E)/(x^2+7)$

$= \frac{A x^{2} (x^{2} + 7) + B x (x^{2} + 7) + C (x^{2} + 7) + x^{3} (D x + E)}{(x^{3}) (x^{2} + 7)}$ $=(Ax^2(x^2+7)+Bx(x^2+7)+C(x^2+7)+x^3(Dx+E))/((x^3)(x^2+7))$

Therefore,

$x^{4} + 6 = A x^{2} (x^{2} + 7) + B x (x^{2} + 7) + C (x^{2} + 7) + x^{3} (D x + E)$ $x^4+6=Ax^2(x^2+7)+Bx(x^2+7)+C(x^2+7)+x^3(Dx+E)$

Let $x = 0$ $x=0$ $\Rightarrow$ $=>$ $6 = 7 C$ $6=7C$ $\Rightarrow$ $=>$ $C = \frac{6}{7}$ $C=6/7$

Coefficients of $x^{2}$ $x^2$ , $\Rightarrow$ $=>$ $0 = 7 A + C$ $0=7A+C$ ; $A = - \frac{6}{49}$ $A=-6/49$

coefficients of $x^{4}$ $x^4$ ; $\Rightarrow$ $=>$ ; $1 = A + D$ $1=A+D$ ; $D = 1 + \frac{6}{49} = \frac{55}{49}$ $D=1+6/49=55/49$

Coefficients of $x^{3}$ $x^3$ ; $\Rightarrow$ $=>$ ; $0 = B + E$ $0=B+E$

Coefficients of $x$ $x$ ; $\Rightarrow$ $=>$ ; $0 = 7 B$ $0=7B$ $\Rightarrow$ $=>$ $B = 0$ $B=0$

$E = 0$ $E=0$

$\frac{x^{4} + 6}{(x^{3}) (x^{2} + 7)} = \frac{- \frac{6}{49}}{x} + \frac{0}{x^{2}} + \frac{\frac{6}{7}}{x^{3}} + \frac{\frac{55}{49} x + 0}{x^{2} + 7}$ $(x^4+6)/((x^3)(x^2+7))=(-6/49)/x+0/x^2+(6/7)/x^3+(55/49x+0)/(x^2+7)$

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{4} + 6}{x^{5} + 7 x^{3}}$ $(x^4+6)/(x^5+7x^3)$ ?

1 Answer

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How do you write the partial fraction decomposition of the rational expression (x^4+6)/(x^5+7x^3)x4+6x5+7x3 (x^4+6)/(x^5+7x^3)?

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How do you write the partial fraction decomposition of the rational expression $\frac{x^{4} + 6}{x^{5} + 7 x^{3}}$ $(x^4+6)/(x^5+7x^3)$ ?