Division of Functions

If [tex]f(x) = x^4 - x^3 + x^2[/tex] and [tex]g(x) = -x^2[/tex], where [tex]x \neq 0[/tex], what is [tex]\frac{f(x)}{g(x)}[/tex]?

A. [tex]x^2 - x + 1[/tex]
B. [tex]x^2 + x + 1[/tex]
C. [tex]-x^2 + x - 1[/tex]
D. [tex]-x^2 - x - 1[/tex]



Answer :

To determine the function [tex]\(\frac{f(x)}{g(x)}\)[/tex] where
[tex]\[ f(x) = x^4 - x^3 + x^2 \][/tex]
and
[tex]\[ g(x) = -x^2 \][/tex]
we need to perform the division of [tex]\(f(x)\)[/tex] by [tex]\(g(x)\)[/tex].

First, write the expression for [tex]\(\frac{f(x)}{g(x)}\)[/tex]:

[tex]\[ r(x) = \frac{f(x)}{g(x)} = \frac{x^4 - x^3 + x^2}{-x^2} \][/tex]

Now, simplify this expression by dividing each term in the numerator by the denominator:

[tex]\[ r(x) = \frac{x^4}{-x^2} - \frac{x^3}{-x^2} + \frac{x^2}{-x^2} \][/tex]

Perform the division for each term:

[tex]\[ r(x) = -x^{4-2} + x^{3-2} - 1 \][/tex]
[tex]\[ r(x) = -x^2 + x - 1 \][/tex]

Thus, the simplified form of the function [tex]\(\frac{f(x)}{g(x)}\)[/tex] is:

[tex]\[ r(x) = -x^2 + x - 1 \][/tex]

So the correct answer is:

[tex]\[ - x^2 + x - 1 \][/tex]