Given that f, of, x, equals, x, squared, minus, 2, x, minus, 24f(x)=x

Question

2
−2x−24 and g, of, x, equals, x, minus, 6g(x)=x−6, find f, of, x, divided by, g, of, xf(x)÷g(x) and express the result as a polynomial in simplest form.

Answer 1

To find \(\frac{f(x)}{g(x)}\), we first need to determine the expressions for \(f(x)\) and \(g(x)\).

Given:

\[ f(x) = x^2 - 2x - 24 \] \[ g(x) = x - 6 \]

Now, we can compute \(\frac{f(x)}{g(x)}\):

\[ \frac{f(x)}{g(x)} = \frac{x^2 - 2x - 24}{x - 6} \]

Next, we can simplify this expression by performing polynomial long division or by factoring the numerator. First, we will factor \(f(x)\):

To factor \(f(x) = x^2 - 2x - 24\), we look for two numbers that multiply to \(-24\) and add to \(-2\). These numbers are \(-6\) and \(4\). Thus, we can factor \(f(x)\) as follows:

\[ f(x) = (x - 6)(x + 4) \]

Now substituting this back into the expression for \(\frac{f(x)}{g(x)}\):

\[ \frac{f(x)}{g(x)} = \frac{(x - 6)(x + 4)}{x - 6} \]

Provided \(x \neq 6\) (to avoid division by zero), we can cancel \(x - 6\) from the numerator and the denominator:

\[ \frac{f(x)}{g(x)} = x + 4 \quad (x \neq 6) \]

Thus, the result of \(\frac{f(x)}{g(x)}\) expressed as a polynomial in simplest form is:

\[ \boxed{x + 4} \]