Question

given that $f(x) = x^2 - 2x - 48$ and $g(x) = x - 8$, find $f(x) \div g(x)$ and express the result as a polynomial in simplest form.
answer
attempt 1 out of 2

Explanation:

Step1: Factor the quadratic function

We start by factoring the quadratic function \( f(x) = x^2 - 2x - 48 \). We need to find two numbers that multiply to \(-48\) and add up to \(-2\). The numbers are \(-8\) and \(6\) because \(-8\times6=-48\) and \(-8 + 6=-2\). So, we can factor \( f(x) \) as follows:

$$ f(x)=x^2 - 2x - 48=(x - 8)(x+6) $$

Step2: Perform the division

Now we need to find \( \frac{f(x)}{g(x)}=\frac{(x - 8)(x + 6)}{x - 8} \). Since \( x
eq8 \) (to avoid division by zero), we can cancel out the common factor \( (x - 8) \) from the numerator and the denominator.

$$ \frac{(x - 8)(x + 6)}{x - 8}=x + 6 $$

Answer:

\( x + 6 \)