Pre-Algebra Examples

$(2 x^{3} - 31 x + 35 - x^{2}) \div (2 x - 7)$

Step 1

Rewrite the division as a fraction.

$\frac{2 x^{3} - 31 x + 35 - x^{2}}{2 x - 7}$

Step 2

Simplify the numerator.

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Step 2.1

Reorder terms.

$\frac{2 x^{3} - x^{2} - 31 x + 35}{2 x - 7}$

Step 2.2

Factor $2 x^{3} - x^{2} - 31 x + 35$ using the rational roots test.

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Step 2.2.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 35, \pm 5, \pm 7$

$q = \pm 1, \pm 2$

Step 2.2.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.5, \pm 35, \pm 17.5, \pm 5, \pm 2.5, \pm 7, \pm 3.5$

Step 2.2.3

Substitute $3.5$ and simplify the expression. In this case, the expression is equal to $0$ so $3.5$ is a root of the polynomial.

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Step 2.2.3.1

Substitute $3.5$ into the polynomial.

$2 \cdot {3.5}^{3} - {3.5}^{2} - 31 \cdot 3.5 + 35$

Step 2.2.3.2

Raise $3.5$ to the power of $3$ .

$2 \cdot 42.875 - {3.5}^{2} - 31 \cdot 3.5 + 35$

Step 2.2.3.3

Multiply $2$ by $42.875$ .

$85.75 - {3.5}^{2} - 31 \cdot 3.5 + 35$

Step 2.2.3.4

Raise $3.5$ to the power of $2$ .

$85.75 - 1 \cdot 12.25 - 31 \cdot 3.5 + 35$

Step 2.2.3.5

Multiply $- 1$ by $12.25$ .

$85.75 - 12.25 - 31 \cdot 3.5 + 35$

Step 2.2.3.6

Subtract $12.25$ from $85.75$ .

$73.5 - 31 \cdot 3.5 + 35$

Step 2.2.3.7

Multiply $- 31$ by $3.5$ .

$73.5 - 108.5 + 35$

Step 2.2.3.8

Subtract $108.5$ from $73.5$ .

$- 35 + 35$

Step 2.2.3.9

Add $- 35$ and $35$ .

$0$

Step 2.2.4

Since $3.5$ is a known root, divide the polynomial by $2 x - 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{2 x^{3} - x^{2} - 31 x + 35}{2 x - 7}$

Step 2.2.5

Divide $2 x^{3} - x^{2} - 31 x + 35$ by $2 x - 7$ .

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Step 2.2.5.1

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

$2 x$

$7$

$2 x^{3}$

$x^{2}$

$31 x$

$35$

Step 2.2.5.2

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

					$x^{2}$
	$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$

Step 2.2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			+	$2 x^{3}$	-	$7 x^{2}$

Step 2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 7 x^{2}$

				$x^{2}$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$

Step 2.2.5.5

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

				$x^{2}$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$

Step 2.2.5.6

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

				$x^{2}$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$

Step 2.2.5.7

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

				$x^{2}$	+	$3 x$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$

Step 2.2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					+	$6 x^{2}$	-	$21 x$

Step 2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 21 x$

				$x^{2}$	+	$3 x$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$

Step 2.2.5.10

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

				$x^{2}$	+	$3 x$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$

Step 2.2.5.11

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

				$x^{2}$	+	$3 x$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$	+	$35$

Step 2.2.5.12

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

				$x^{2}$	+	$3 x$	-	$5$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$	+	$35$

Step 2.2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	-	$5$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$	+	$35$
							-	$10 x$	+	$35$

Step 2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x + 35$

				$x^{2}$	+	$3 x$	-	$5$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$	+	$35$
							+	$10 x$	-	$35$

Step 2.2.5.15

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

				$x^{2}$	+	$3 x$	-	$5$
$2 x$	-	$7$		$2 x^{3}$	-	$x^{2}$	-	$31 x$	+	$35$
			-	$2 x^{3}$	+	$7 x^{2}$
					+	$6 x^{2}$	-	$31 x$
					-	$6 x^{2}$	+	$21 x$
							-	$10 x$	+	$35$
							+	$10 x$	-	$35$
										$0$

Step 2.2.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 3 x - 5$

Step 2.2.6

Write $2 x^{3} - x^{2} - 31 x + 35$ as a set of factors.

$\frac{(2 x - 7) (x^{2} + 3 x - 5)}{2 x - 7}$

Step 3

Cancel the common factor of $2 x - 7$ .

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Step 3.1

Cancel the common factor.

$\frac{(2 x - 7) (x^{2} + 3 x - 5)}{2 x - 7}$

Step 3.2

Divide $x^{2} + 3 x - 5$ by $1$ .

$x^{2} + 3 x - 5$