Basic Math Examples

$(- 9 q^{3} - 11 q - 140) \div (3 q + 7)$

Step 1

Factor $- 1$ out of $- 9 q^{3} - 11 q - 140$ .

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Step 1.1

Factor $- 1$ out of $- 9 q^{3}$ .

$(- (9 q^{3}) - 11 q - 140) \div (3 q + 7)$

Step 1.2

Factor $- 1$ out of $- 11 q$ .

$(- (9 q^{3}) - (11 q) - 140) \div (3 q + 7)$

Step 1.3

Rewrite $- 140$ as $- 1 (140)$ .

$(- (9 q^{3}) - (11 q) - 1 (140)) \div (3 q + 7)$

Step 1.4

Factor $- 1$ out of $- (9 q^{3}) - (11 q)$ .

$(- (9 q^{3} + 11 q) - 1 (140)) \div (3 q + 7)$

Step 1.5

Factor $- 1$ out of $- (9 q^{3} + 11 q) - 1 (140)$ .

$- (9 q^{3} + 11 q + 140) \div (3 q + 7)$

Step 2

Factor.

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Step 2.1

Factor $9 q^{3} + 11 q + 140$ using the rational roots test.

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Step 2.1.1

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.

$p = \pm 1, \pm 140, \pm 2, \pm 70, \pm 4, \pm 35, \pm 5, \pm 28, \pm 7, \pm 20, \pm 10, \pm 14$

$q = \pm 1, \pm 9, \pm 3$

Step 2.1.2

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

$\pm 1, \pm 0.\overline{1}, \pm 0.\overline{3}, \pm 140, \pm 15.\overline{5}, \pm 46.\overline{6}, \pm 2, \pm 0.\overline{2}, \pm 0.\overline{6}, \pm 70, \pm 7.\overline{7}, \pm 23.\overline{3}, \pm 4, \pm 0.\overline{4}, \pm 1.\overline{3}, \pm 35, \pm 3.\overline{8}, \pm 11.\overline{6}, \pm 5, \pm 0.\overline{5}, \pm 1.\overline{6}, \pm 28, \pm 3.\overline{1}, \pm 9.\overline{3}, \pm 7, \pm 0.\overline{7}, \pm 2.\overline{3}, \pm 20, \pm 2.\overline{2}, \pm 6.\overline{6}, \pm 10, \pm 1.\overline{1}, \pm 3.\overline{3}, \pm 14, \pm 1.\overline{5}, \pm 4.\overline{6}$

Step 2.1.3

Substitute $- 2.\overline{3}$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2.\overline{3}$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $- 2.\overline{3}$ into the polynomial.

$9 {(- 2.\overline{3})}^{3} + 11 \cdot - 2.\overline{3} + 140$

Step 2.1.3.2

Raise $- 2.\overline{3}$ to the power of $3$ .

$9 \cdot - 12.\overline{703} + 11 \cdot - 2.\overline{3} + 140$

Step 2.1.3.3

Multiply $9$ by $- 12.\overline{703}$ .

$- 114.\overline{3} + 11 \cdot - 2.\overline{3} + 140$

Step 2.1.3.4

Multiply $11$ by $- 2.\overline{3}$ .

$- 114.\overline{3} - 25.\overline{6} + 140$

Step 2.1.3.5

Subtract $25.\overline{6}$ from $- 114.\overline{3}$ .

$- 140 + 140$

Step 2.1.3.6

Add $- 140$ and $140$ .

$0$

Step 2.1.4

Since $- 2.\overline{3}$ is a known root, divide the polynomial by $3 q + 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 q^{3} + 11 q + 140}{3 q + 7}$

Step 2.1.5

Divide $9 q^{3} + 11 q + 140$ by $3 q + 7$ .

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Step 2.1.5.1

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

$3 q$

$7$

$9 q^{3}$

$0 q^{2}$

$11 q$

$140$

Step 2.1.5.2

Divide the highest order term in the dividend $9 q^{3}$ by the highest order term in divisor $3 q$ .

					$3 q^{2}$
	$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$3 q^{2}$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			+	$9 q^{3}$	+	$21 q^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $9 q^{3} + 21 q^{2}$

				$3 q^{2}$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$

Step 2.1.5.5

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

				$3 q^{2}$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$

Step 2.1.5.6

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

				$3 q^{2}$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$

Step 2.1.5.7

Divide the highest order term in the dividend $- 21 q^{2}$ by the highest order term in divisor $3 q$ .

				$3 q^{2}$	-	$7 q$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$3 q^{2}$	-	$7 q$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					-	$21 q^{2}$	-	$49 q$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 21 q^{2} - 49 q$

				$3 q^{2}$	-	$7 q$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$

Step 2.1.5.10

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

				$3 q^{2}$	-	$7 q$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$

Step 2.1.5.11

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

				$3 q^{2}$	-	$7 q$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$	+	$140$

Step 2.1.5.12

Divide the highest order term in the dividend $60 q$ by the highest order term in divisor $3 q$ .

				$3 q^{2}$	-	$7 q$	+	$20$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$	+	$140$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$3 q^{2}$	-	$7 q$	+	$20$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$	+	$140$
							+	$60 q$	+	$140$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $60 q + 140$

				$3 q^{2}$	-	$7 q$	+	$20$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$	+	$140$
							-	$60 q$	-	$140$

Step 2.1.5.15

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

				$3 q^{2}$	-	$7 q$	+	$20$
$3 q$	+	$7$		$9 q^{3}$	+	$0 q^{2}$	+	$11 q$	+	$140$
			-	$9 q^{3}$	-	$21 q^{2}$
					-	$21 q^{2}$	+	$11 q$
					+	$21 q^{2}$	+	$49 q$
							+	$60 q$	+	$140$
							-	$60 q$	-	$140$
										$0$

Step 2.1.5.16

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

$3 q^{2} - 7 q + 20$

Step 2.1.6

Write $9 q^{3} + 11 q + 140$ as a set of factors.

$- ((3 q + 7) (3 q^{2} - 7 q + 20)) \div (3 q + 7)$

Step 2.2

Remove unnecessary parentheses.

$- (3 q + 7) (3 q^{2} - 7 q + 20) \div (3 q + 7)$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Reduce the expression $- (3 q + 7) (3 q^{2} - 7 q + 20) \div (3 q + 7)$ by cancelling the common factors.

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Step 3.1.1

Cancel the common factor.

$- (3 q + 7) (3 q^{2} - 7 q + 20) \div (3 q + 7)$

Step 3.1.2

Rewrite the expression.

$\frac{- 1 (3 q^{2} - 7 q + 20)}{1}$

Step 3.2

Divide $- 1 (3 q^{2} - 7 q + 20)$ by $1$ .

$- 1 (3 q^{2} - 7 q + 20)$