Question

when combining like terms, do i do 3x^2 plus negative 2x^2? or does the sign change to addition, 3x^2 plus 2x^2?

Answer 1

Let's start using this property:

$\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}$

In this case:

$\begin{gathered} a=3x^2-2x-1 \\ c=-2x^2+x-5 \\ b=x-5 \\ so\colon \\ \frac{3x^2-2x-1}{x-5}+\frac{-2x^2+x-5}{x-5}=\frac{(3x^2-2x-1)+(-2x^2+x-5)}{x-5} \end{gathered}$

Now, let's add like terms in the numerator:

$\begin{gathered} \frac{3x^2-2x-1-2x^2+x-5}{x-5}=\frac{(3x^2-2x^2)+(-2x+x)+(-1-5)}{x-5} \\ \to\frac{x^2-x-6}{x-5} \end{gathered}$

For the numerator:

$x^2-x-6$

The factors of -6 that sum to -1 are 2 and -3, therefore:

$x^2-x-6=(x+2)(x-3)$

Hence:

$\frac{x^2-x-6}{x-5}=\frac{(x+2)(x-3)}{x-5}$