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Precalculus 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

More step-by-step Precalculus examples from this chapter.

Solving for a Variable Solving by Graphing Finding the Remainder Using Long Polynomial Division Synthetic Division Factoring Using Any Method Factoring a Difference of Squares Factoring a Sum of Cubes Factoring Out Greatest Common Factor (GCF) Factoring by Grouping Factoring Trinomials Partial Fraction Decomposition Finding the Perfect Square Trinomial

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