Question

How do I write the partial fraction decomposition of (x^2)/(x^2+x+5)

Answer 1

I don't get it

Answer 2

you cannot factor the denominator with real numbers
divide first

Answer 3

the degree of the numerator is the same as the degree of the denominator
you have to divide first, then try partial fractions

Answer 4

or maybe use this trick
\[\frac{x^2}{x^2+x+5}=\frac{x^2+x+5-x-5}{x^2+x+5}=1+\frac{-x-5}{x^2+x+5}\]

Answer 5

but the denominator does not factor over the integers, you can check because it has no real zeros

Answer 6

so do i write it like A + (-Bx-C)/(x^2+x+5) ?

Answer 7

it is already done above

Answer 8

I dont have to use A, B, C ,etc??

Answer 9

no look at the gimmick i wrote

Answer 10

you have \(x^2\) up top and \(x^2\) in the denominator so you know it is going to be \(1+stuff\)

Answer 11

how do you write it with letters. i have to do it that way

Answer 12

i am not sure what your end goal is here
usually this is a step on the way to integration, but not here

Answer 13

i don't know what you mean by "letters" you can do this by division, not really much else to do, but clearly the simple trick is much easier than dividing