To solve the problem of finding [tex]\((f / g)(x)\)[/tex] given the functions [tex]\(f(x) = x^4 - x^3 + x^2\)[/tex] and [tex]\(g(x) = -x^2\)[/tex], where [tex]\(x \neq 0\)[/tex], we need to divide [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex].
Here’s the step-by-step solution:
1. Write the quotient [tex]\((f / g)(x)\)[/tex]: The goal is to divide the function [tex]\(f(x)\)[/tex] by the function [tex]\(g(x)\)[/tex].
[tex]\[
(f / g)(x) = \frac{f(x)}{g(x)}
\][/tex]
2. Substitute the given functions: Substitute [tex]\(f(x) = x^4 - x^3 + x^2\)[/tex] and [tex]\(g(x) = -x^2\)[/tex] into the quotient.
[tex]\[
(f / g)(x) = \frac{x^4 - x^3 + x^2}{-x^2}
\][/tex]
3. Simplify the expression: Divide each term in the numerator [tex]\(x^4 - x^3 + x^2\)[/tex] by the denominator [tex]\(-x^2\)[/tex].
- Divide [tex]\(x^4\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{x^4}{-x^2} = -x^2
\][/tex]
- Divide [tex]\(x^3\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{x^3}{-x^2} = -x
\][/tex]
- Divide [tex]\(x^2\)[/tex] by [tex]\(-x^2\)[/tex]:
[tex]\[
\frac{x^2}{-x^2} = -1
\][/tex]
Now, combine these results:
[tex]\[
(f / g)(x) = -x^2 - x - 1
\][/tex]
Therefore, the simplified form of [tex]\((f / g)(x)\)[/tex] is [tex]\(-x^2 - x - 1\)[/tex].
So the correct answer is:
[tex]\[
- x^2 - x - 1
\][/tex]
Hence, the solution is:
[tex]\(-x^2 - x - 1\)[/tex].
