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

.

\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 $x^2 + x - 2$ , which are $1$ , $1$ , and $-2$ . Then write the root of $x + 2$ , which is $-2$ , to the left.
Bring down leading coefficient: Bring down the leading coefficient, which is $1$ .
Multiply and add coefficients: Multiply the root $-2$ by the leading coefficient $1$ and write the result under the second coefficient.
Continue with multiplication and addition: Add the second coefficient $1$ and the result of the multiplication $-2 \times 1$ , which is $-2$ . Write the sum, which is $-1$ , under the line.
Check for exact division: Multiply the root $-2$ by the new number $-1$ and write the result, which is $2$ , under the third coefficient.
Check for exact division: Multiply the root $-2$ by the new number $-1$ and write the result, which is $2$ , under the third coefficient.Add the third coefficient $-2$ and the result of the multiplication $-2 \times -1$ , which is $2$ . Write the sum, which is $0$ , under the line.
Check for exact division: Multiply the root $-2$ by the new number $-1$ and write the result, which is $2$ , under the third coefficient.Add the third coefficient $-2$ and the result of the multiplication $-2 \times -1$ , which is $2$ . Write the sum, which is $0$ , under the line.The numbers below the line represent the coefficients of the quotient polynomial $q(x)$ . Since there is no remainder, $r$ is $0$ and the division is exact.

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