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Use synthetic division to find $(x^2 - 4x + 34) \div (x - 1)$ . $\newline$ 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 - 4x + 34) \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 with $1$ as the zero from $(x - 1)$ and the coefficients of the polynomial $x^2 - 4x + 34$ , which are $1$ , $-4$ , and $34$ .
Bring down leading coefficient: Bring down the leading coefficient, which is $1$ .
Multiply zero by leading coefficient: Multiply the zero ( $1$ ) by the leading coefficient ( $1$ ) and write the result under the next coefficient ( $-4$ ).
Add numbers in second column: Add the numbers in the second column: $-4 + (1 \times 1) = -3$ .
Multiply zero by number obtained: Multiply the zero ( $1$ ) by the number just obtained ( $-3$ ) and write the result under the next coefficient ( $34$ ).
Add numbers in third column: Add the numbers in the third column: $34 + (1 \times -3) = 31$ .
Write result of synthetic division: Write the result of synthetic division as a polynomial $q(x)$ plus the remainder over the divisor: $q(x) = x - 3$ and the remainder is $31$ .
Express final answer: Express the final answer in the form $q(x) + \frac{r}{d(x)}$ : $(x - 3) + \frac{31}{(x - 1)}$ .

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