Answer
$\frac{5f(x)}{g(x)} = 5f(x) \div g(x) = \dfrac{10x + 25}{x^2 - 3x + 2}$
Domain: all real numbers except for $2$ or $1$
Work Step by Step
This exercise asks us to divide $5$ times $f(x)$ by $g(x)$. Let's write out the problem:
$\frac{5f(x)}{g(x)} = 5f(x) \div g(x) = \frac{5(2x + 5)}{x^2 - 3x + 2}$
Distribute terms in the numerator:
$\frac{5f(x)}{g(x)} = 5f(x) \div g(x) = \frac{10x + 25}{x^2 - 3x + 2}$
When we find the domain, we want to find which values of $x$ will cause the function to become undefined; in other words, we want to find any restrictions for $x$. We set the denominator equal to $0$ and solve for x.
We have a quadratic expression in the denominator, which is in the form $ax^2 + bx + c$. We need to find which factors multiplied together will equal $ac$ but when added together will equal $b$.
In this equation, $ac$ is $2$ and $b$ is $-3$. The factors $-2$ and $-1$ will work. Let's write the expression in factor form:
$(x - 2)(x - 1)$
Let's set the factors equal to $0$:
$(x - 2)(x - 1) = 0$
Set the first factor equal to $0$:
$x - 2 = 0$
Add $2$ to each side of the equation to solve for $x$:
$x = 2$
Set the other factor equal to $0$:
$x - 1 = 0$
Add $1$ to each side of the equation:
$x = 1$
In this exercise, $x$ can be any real number except for $2$ and $1$.
