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Use synthetic division to find $(x^2 + 12x + 11) \div (x + 1)$ . $\newline$ 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. $\newline$ _________

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Divide polynomials using synthetic division

Full solution

Q. Use synthetic division to find

(x^2 + 12x + 11) \div (x + 1)

.

\newline

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.

\newline

_________

Set up synthetic division: Set up synthetic division by writing the coefficients of the dividend ( $1$ for $x^2$ , $12$ for $x$ , and $11$ for the constant term) and the root of the divisor, which is $-1$ for $(x + 1)$ .
Perform synthetic division: Perform synthetic division: $\newline$ $-1$ | $1$ $12$ $11$ $\newline$ | $-1$ $-11$ $\newline$ ------------ $\newline$ $1$ $11$ $0$
Write result of synthetic division: Write the result of synthetic division as a polynomial plus remainder over divisor: $q(x) = 1x + 11$ and the remainder $r = 0$ .
Check for remainder: Since the remainder is $0$ , the result is just the polynomial $q(x)$ without any remainder term.

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