Question

Write it in reduced form as a ratio of polynomials p(x)/q(x)

Answer 1

We are given the following expression

`(x^2)/(x-5)-(8)/(x-2)`

Let us re-write the expression as a ratio of polynomials p(x)/q(x)

First of all, find the least common multiple (LCM) of the denominators.

The LCM of the denominators is given by

`(x-5)(x-2)`

Now, adjust the fractions based on the LCM

`\begin{gathered} (x^2)/(x-5)*((x-2))/((x-2))=(x^2(x-2))/((x-5)(x-2)) \\ \frac{8^{}}{x-2}*((x-5))/((x-5))=(8(x-5))/((x-2)(x-5)) \end{gathered}`

So, the expression becomes

`(x^2(x-2))/((x-5)(x-2))-(8(x-5))/((x-2)(x-5))`

Now, apply the fraction rule

`(a)/(c)-(b)/(c)=(a-b)/(c)`
`(x^2(x-2))/((x-5)(x-2))-(8(x-5))/((x-2)(x-5))=(x^2(x-2)-8(x-5))/((x-5)(x-2))`

Finally, expand the products in the numerator

`(x^2(x-2)-8(x-5))/((x-5)(x-2))=(x^3-2x^2-8x+40)/((x-5)(x-2))`

Therefore, the given expression as a ratio of polynomials p(x)/q(x) is

`(x^3-2x^2-8x+40)/((x-5)(x-2))`