Divide Using Long Polynomial Division (2x^3-10x^2+16x-5)÷(x^2-6x+8)

$(2 x^{3} - 10 x^{2} + 16 x - 5) \div (x^{2} - 6 x + 8)$

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

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

Divide the highest order term in the dividend $2 x^{3}$ by the highest order term in divisor $x^{2}$ .

$2 x$

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

Multiply the new quotient term by the divisor.

$2 x$

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

$2 x^{3}$

$12 x^{2}$

$16 x$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 12 x^{2} + 16 x$

$2 x$

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

$2 x^{3}$

$12 x^{2}$

$16 x$

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

$2 x$

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

$2 x^{3}$

$12 x^{2}$

$16 x$

$2 x^{2}$

$0$

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

						$2 x$
$x^{2}$	-	$6 x$	+	$8$		$2 x^{3}$	-	$10 x^{2}$	+	$16 x$	-	$5$
					-	$2 x^{3}$	+	$12 x^{2}$	-	$16 x$
							+	$2 x^{2}$	+	$0$	-	$5$

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

						$2 x$	+	$2$
$x^{2}$	-	$6 x$	+	$8$		$2 x^{3}$	-	$10 x^{2}$	+	$16 x$	-	$5$
					-	$2 x^{3}$	+	$12 x^{2}$	-	$16 x$
							+	$2 x^{2}$	+	$0$	-	$5$

Multiply the new quotient term by the divisor.

$2 x$

$2$

$x^{2}$

$6 x$

$8$

$2 x^{3}$

$10 x^{2}$

$16 x$

$5$

$2 x^{3}$

$12 x^{2}$

$16 x$

$2 x^{2}$

$0$

$5$

$2 x^{2}$

$12 x$

$16$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 12 x + 16$

						$2 x$	+	$2$
$x^{2}$	-	$6 x$	+	$8$		$2 x^{3}$	-	$10 x^{2}$	+	$16 x$	-	$5$
					-	$2 x^{3}$	+	$12 x^{2}$	-	$16 x$
							+	$2 x^{2}$	+	$0$	-	$5$
							-	$2 x^{2}$	+	$12 x$	-	$16$

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

						$2 x$	+	$2$
$x^{2}$	-	$6 x$	+	$8$		$2 x^{3}$	-	$10 x^{2}$	+	$16 x$	-	$5$
					-	$2 x^{3}$	+	$12 x^{2}$	-	$16 x$
							+	$2 x^{2}$	+	$0$	-	$5$
							-	$2 x^{2}$	+	$12 x$	-	$16$
									+	$12 x$	-	$21$

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

$2 x + 2 + \frac{12 x - 21}{x^{2} - 6 x + 8}$

Divide Using Long Polynomial Division (2x^3-10x^2+16x-5)÷(x^2-6x+8)

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