Question

use synthetic division to simplify \\(\frac{x^2 - 3}{x + 3}\\).
write your answer in the form \\(q(x) + \frac{r}{d(x)}\\), where \\(q(x)\\) is a polynomial, \\(r\\) is an integer, and \\(d(x)\\) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify divisor root

Divisor is $x + 3$, so root is $-3$.

Step2: List coefficients

Dividend $x^2 - 3$ has coefficients $1, 0, -3$.

Step3: Perform synthetic division

Bring down 1. Multiply by $-3$: $1×(-3) = -3$. Add to next coefficient: $0 + (-3) = -3$. Multiply by $-3$: $-3×(-3) = 9$. Add to last coefficient: $-3 + 9 = 6$.

Step4: Form result

Quotient $q(x) = x - 3$, remainder $r = 6$, divisor $d(x) = x + 3$. So $\frac{x^2 - 3}{x + 3} = x - 3 + \frac{6}{x + 3}$.

Answer:

$x - 3 + \frac{6}{x + 3}$