Answer
$-f(x) \cdot g(x) = -x^3 + 5x^2 - 8x + 4$
$\text{The domain is all real numbers.}$
Work Step by Step
In this problem, we are asked to multiply two functions. Let us go ahead and set up the multiplication problem, separating the two functions with parentheses:
$-f(x) \cdot g(x) = -(x - 2)(x^2 - 3x + 2)$
Let's distribute the $-$ sign to each of the terms in the binomial to get:
$-f(x) \cdot g(x) = (-x + 2)(x^2 - 3x + 2)$
Multiply each term in the binomial with each term in the trinomial:
$-f(x) \cdot g(x) = (-x)(x^2) + (2)(x^2) + (-3x)(-x) + (-3x)(2) + (2)(-x) + (2)(2)$
Multiply out, to simplify:
$-f(x) \cdot g(x) = -x^3 + 2x^2 + 3x^2 - 6x - 2x + 4$
Group like terms:
$-f(x) \cdot g(x) = -x^3 + (2x^2 + 3x^2) + (-6x - 2x) + 4$
Combine like terms:
$-f(x) \cdot g(x) = -x^3 + 5x^2 - 8x + 4$
There are no restrictions on $x$.
The domain is the set of all real numbers since there are no values of $x$ that will make the function undefined.
