Question

use synthetic division to simplify \\(\frac{x^{5} + 16x^{4} + 33x^{3} + 18x^{2}}{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 coefficients and root

Coefficients of dividend: $1, 16, 33, 18, 0, 0$ (for missing $x$ and constant terms); root: $-1$ (from $x+1=0$)

Step2: Apply synthetic division

Bring down 1. Multiply by -1: $-1$. Add to 16: $15$.
Multiply 15 by -1: $-15$. Add to 33: $18$.
Multiply 18 by -1: $-18$. Add to 18: $0$.
Multiply 0 by -1: $0$. Add to 0: $0$.
Multiply 0 by -1: $0$. Add to 0: $0$.

Step3: Form quotient and remainder

Quotient: $x^4 + 15x^3 + 18x^2 + 0x + 0$; Remainder: $0$

Step4: Write final form

Simplify to $x^4 + 15x^3 + 18x^2 + \frac{0}{x+1}$

Answer:

$x^4 + 15x^3 + 18x^2 + \frac{0}{x+1}$