Question

use synthetic division to simplify \\(\frac{x^2 + 3x}{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

Set $x+3=0$, so root is $-3$.

Step2: List coefficients

Dividend $x^2+3x$ has coefficients $[1,3,0]$.

Step3: Apply synthetic division

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

Step4: Write result

Quotient $q(x)=x$, remainder $r=0$. Form: $x + \frac{0}{x+3}$.

Answer:

$x + \frac{0}{x+3}$