How do you write the partial fraction decomposition of the rational expression $[((3x^2)+3x+12) / ((x-5)(x^2+9))]$ ?

Question

1370 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-27T10:39:52

The answer is $=3/(x-5)+3/(x^2+9)$

Let's work out the decomposition into partial fractions

$(3x^2+3x+12)/((x-5)(x^2+9))=A/(x-5)+(Bx+C)/(x^2+9)$

$=(A(x^2+9)+(Bx+C)(x-5))/((x-5)(x^2+9))$

Since the denominators are the same, we can compare the numerators

$3x^2+3x+12=A(x^2+9)+(Bx+C)(x-5)$

Let $x=5$ , $=>$ , $75+15+12=(25+9)A$

$=>$ , $34A=102$ , $=>$ , $A=3$

Coefficients of $x^2$ , $3=A+B$

$=>$ , $B=3-A=0$

Coefficients of $x$ , $=>$ , $3=C$

So,

$(3x^2+3x+12)/((x-5)(x^2+9))=3/(x-5)+3/(x^2+9)$

How do you write the partial fraction decomposition of the rational expression [((3x^2)+3x+12) / ((x-5)(x^2+9))][((3x^2)+3x+12) / ((x-5)(x^2+9))]?