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Use synthetic division to find $(x^2 + 2x + 7) \div (x + 5)$ . $\newline$ 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 + 2x + 7) \div (x + 5)

.

\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 with $-5$ (the root of $x + 5$ ) and the coefficients of the polynomial $x^2 + 2x + 7$ , which are $1$ , $2$ , and $7$ .
Perform synthetic division: Perform synthetic division: $\newline$ $-5$ | $1$ $2$ $7$ $\newline$ | $-5$ $-15$ $\newline$ ------------ $\newline$ $1$ $-3$ $-8$
Obtain quotient and remainder: The result of the synthetic division gives us the coefficients of the quotient polynomial and the remainder. The quotient is $x - 3$ and the remainder is $-8$ .
Write final answer: Write the answer in the form $q(x) + \frac{r}{d(x)}$ . The quotient polynomial $q(x)$ is $x - 3$ , the remainder $r$ is $-8$ , and the divisor $d(x)$ is $x + 5$ .

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