Question

rewrite the rational expression \\(\frac{x^2 - 9}{x - 3}\\) using long division and express it in the form \\(q(x) + \frac{r(x)}{b(x)}\\). fill in the blanks. use the keyboard to type your answer in each box. after performing long division, the quotient \\(q(x)\\) is \\(\square\\) and the remainder \\(r(x)\\) is \\(\square\\). therefore, the expression in its simplified form is \\(\square\\).

Explanation:

Step1: Factor numerator (optional check)

$x^2 - 9 = (x-3)(x+3)$

Step2: Perform polynomial long division

Divide $x^2$ by $x$: $x$. Multiply $(x-3)$ by $x$: $x^2 - 3x$. Subtract from dividend:
$(x^2 - 9) - (x^2 - 3x) = 3x - 9$

Step3: Divide remaining term

Divide $3x$ by $x$: $3$. Multiply $(x-3)$ by $3$: $3x - 9$. Subtract:
$(3x - 9) - (3x - 9) = 0$

Step4: Identify quotient and remainder

Quotient $q(x) = x+3$, remainder $r(x)=0$

Answer:

After performing long division, the quotient $q(x)$ is $x+3$ and the remainder $r(x)$ is $0$. Therefore, the expression in its simplified form is $x+3$.