Question2. Rewrite $g(x)=\frac{-5 x^{3}-18}{x^{3}+4}$ in the form $g(x)=q(x)+\frac{r}{b(x)}$ .

Studdy Solution

STEP 1

1. We are given a rational function $g(x) = \frac{-5x^3 - 18}{x^3 + 4}$ .
2. We need to express $g(x)$ in the form $g(x) = q(x) + \frac{r}{b(x)}$ , where $q(x)$ is a polynomial, $r$ is a constant, and $b(x)$ is the denominator of the original function.

STEP 2

1. Perform polynomial long division to divide the numerator by the denominator.
2. Express the result in the desired form.

STEP 3

Perform polynomial long division of $-5x^3 - 18$ by $x^3 + 4$ .
- Divide the leading term of the numerator, $-5x^3$ , by the leading term of the denominator, $x^3$ , to get $-5$ . - Multiply the entire divisor $x^3 + 4$ by $-5$ to get $-5x^3 - 20$ . - Subtract this result from the original numerator $-5x^3 - 18$ :
$(-5x^3 - 18) - (-5x^3 - 20) = 2$

STEP 4

Express $g(x)$ in the form $g(x) = q(x) + \frac{r}{b(x)}$ .
- The quotient from the division is $-5$ , and the remainder is $2$ . - Therefore, $g(x) = -5 + \frac{2}{x^3 + 4}$ .
Thus, the expression $g(x)$ in the desired form is:
$g(x) = -5 + \frac{2}{x^3 + 4}$

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