Find the Remainder 8x^2-6x+6 , 2x+1

$8 x^{2} - 6 x + 6$ , $2 x + 1$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

$\frac{8 x^{2} - 6 x + 6}{2 x + 1}$

Step 2

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$8 x^{2}$

$6 x$

$6$

Step 3

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $2 x$ .

					$4 x$
	$2 x$	+	$1$		$8 x^{2}$	-	$6 x$	+	$6$

Step 4

Multiply the new quotient term by the divisor.

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} + 4 x$

Step 6

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Step 7

Pull the next terms from the original dividend down into the current dividend.

Step 8

Divide the highest order term in the dividend $- 10 x$ by the highest order term in divisor $2 x$ .

Step 9

Multiply the new quotient term by the divisor.

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x - 5$

Step 11

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

Step 12

The final answer is the quotient plus the remainder over the divisor.

$4 x - 5 + \frac{11}{2 x + 1}$

Step 13

The remainder is the part of the answer that is left after the division by $2 x + 1$ is complete.

$11$