Question

given that $f(x) = x^2 - 12x + 35$ and $g(x) = x - 7$, find $(f \div g)(x)$ and express the result as a polynomial in simplest form.

answer
attempt 1 out of 2

submit answer

Explanation:

Step1: Factor the quadratic function

We start by factoring the quadratic function \( f(x) = x^2 - 12x + 35 \). We need to find two numbers that multiply to \( 35 \) and add up to \( -12 \). The numbers \( -5 \) and \( -7 \) work because \( (-5)\times(-7)=35 \) and \( -5 + (-7)=-12 \). So, we can factor \( f(x) \) as follows:

$$ f(x)=x^2 - 12x + 35=(x - 5)(x - 7) $$

Step2: Perform the division \((f\div g)(x)\)

By the definition of function division, \((f\div g)(x)=\frac{f(x)}{g(x)}\) (where \( g(x)
eq0 \)). We know that \( f(x)=(x - 5)(x - 7) \) and \( g(x)=x - 7 \) (and \( x
eq7 \) to avoid division by zero). So we substitute these into the division formula:

$$ \frac{f(x)}{g(x)}=\frac{(x - 5)(x - 7)}{x - 7} $$

Since \( x
eq7 \), we can cancel out the common factor of \( x - 7 \) in the numerator and the denominator. This leaves us with:

$$ \frac{(x - 5)(x - 7)}{x - 7}=x - 5 $$

Answer:

\( x - 5 \) (with the restriction \( x
eq7 \))