How Do You Factor the Denominator of a Rational Function for Partial Fractions?

[iamtheman]

Can someone pls help me solve this integral?

integral of (6x^2-13x-43)dx/(x^3-1x^2-8x+12)

it's supposed to be solved using partial fractions, but I am having trouble factoring the denom correctly so I can apply it...

Thanks

 

[matt grime]

2 appears to be a root of the denominator, but why did you specify the coeff of the x^2 term? it appears to be a typo because of it.

 

[Hurkyl]

it's supposed to be solved using partial fractions, but I am having trouble factoring the denom correctly so I can apply it...

There's a very useful theorem about rational roots of an integer polynomial:


Theorem: If $y$ is a rational number that is a root of the polynomial

$$f(x) = a_0 + a_1 x + \ldots + a_n x^n \quad (a_n \neq 0)$$

Then $y$ can be written as $p / q$ for some integers $p$ and $q$ where $p | a_0$ and $q | a_n$.

($a | b$ means "a divides b")


Using this theorem, if $x^3 - x^2 - 8x + 12$ has a rational root, then it can be written in the form $p/q$ where $p \in \{1, -1, 2, -2, 3, -3, 4, -4, 6, -6, 12, -12\}$ and $q \in \{1, -1\}$. Only 12 possibilities to try, so if one exists you can find it by exhaustion.

When trying to factor large polynomials, this is usually a good place to start.