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Calculus Examples Algebra Concepts and Expressions

Divide Using Long Polynomial Division

$\frac{6 x^{2} + 5 x - 7}{x + 2}$

Step 1

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

$x$

+

$2$

$6 x^{2}$

+

$5 x$

-

$7$

Step 2

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

					$6 x$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$

Step 3

Multiply the new quotient term by the divisor.

					$6 x$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				+	$6 x^{2}$	+	$12 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 12 x$

					$6 x$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$

Step 5

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

					$6 x$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$

Step 6

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

					$6 x$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$	-	$7$

Step 7

Divide the highest order term in the dividend $- 7 x$ by the highest order term in divisor $x$ .

					$6 x$	-	$7$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$	-	$7$

Step 8

Multiply the new quotient term by the divisor.

					$6 x$	-	$7$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$	-	$7$
						-	$7 x$	-	$14$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x - 14$

					$6 x$	-	$7$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$	-	$7$
						+	$7 x$	+	$14$

Step 10

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

					$6 x$	-	$7$
	$x$	+	$2$		$6 x^{2}$	+	$5 x$	-	$7$
				-	$6 x^{2}$	-	$12 x$
						-	$7 x$	-	$7$
						+	$7 x$	+	$14$
								+	$7$

Step 11

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

$6 x - 7 + \frac{7}{x + 2}$

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Algebra Concepts and Expressions

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Solving for a Variable Synthetic Division Partial Fraction Decomposition Dividing Finding the Domain Finding the Range Finding the Domain and Range Finding the Asymptotes Quadratic Inequalities Rational Inequalities Converting from Interval to Inequality Finding the Holes in a Graph

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$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$