Question

use synthetic division to simplify \\(\frac{x^3 + 8x^2 + 7x}{x + 1}\\).
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 root of divisor

Divisor $d(x)=x+1$, root is $-1$.

Step2: List coefficients of dividend

Dividend $x^3+8x^2+7x+0$, coefficients: $1,8,7,0$.

Step3: Perform synthetic division

Bring down 1. Multiply by $-1$: $-1$. Add to 8: $7$. Multiply by $-1$: $-7$. Add to7: $0$. Multiply by $-1$: $0$. Add to0: $0$.

Step4: Form quotient and remainder

Quotient: $x^2+7x$, remainder: $0$. Result: $x^2+7x+\frac{0}{x+1}=x^2+7x$.

Answer:

$x^2 + 7x + 0$