Question

How do I convert the following expression into a rational function p(x)/q(x), where p(x) and q(x) are polynomials?

r(x)=(3/(x+2))-((2x-4)/( `x^2` -4))+(x/(x-2))

Answer 1

`r(x)=(x^2+3x-2)/(x^2-4)`.
`p(x)=x^2+3x-2` and `q(x)=x^2-4`

Explanation:

When converting this into a rational function in the form p(x)/q(x), you are essentially just combining all of the terms of r(x) into one big fraction. This means that the first step is to identify the least common denominator of the terms of r(x), which in this case is `x^2-4`. Next, you must get all of the terms to have that denominator:

`r(x)=(3/(x+2)xx(x-2)/(x-2))-(2x-4)/(x^2-4)+(x/(x-2)xx(x+2)/(x+2))`

`r(x)=(3(x-2)-(2x-4)+x(x+2))/(x^2-4)`

From here, you can expand and simplify.

`r(x)=(3x-6-2x+4+x^2+2x)/(x^2-4)`
`r(x)=(x^2+3x-2)/(x^2-4)`

Now that this is simplified, we can see that the numerator, p(x), is equal to `x^2+3x-2` and the denominator, q(x), is equal to `x^2-4`