Algebra Examples

$9 x^{3} - 6 x^{2} + 12 x^{5} - 18 x^{7}$

Step 1

Factor $3 x^{2}$ out of $9 x^{3} - 6 x^{2} + 12 x^{5} - 18 x^{7}$ .

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Step 1.1

Factor $3 x^{2}$ out of $9 x^{3}$ .

$3 x^{2} (3 x) - 6 x^{2} + 12 x^{5} - 18 x^{7}$

Step 1.2

Factor $3 x^{2}$ out of $- 6 x^{2}$ .

$3 x^{2} (3 x) + 3 x^{2} (- 2) + 12 x^{5} - 18 x^{7}$

Step 1.3

Factor $3 x^{2}$ out of $12 x^{5}$ .

$3 x^{2} (3 x) + 3 x^{2} (- 2) + 3 x^{2} (4 x^{3}) - 18 x^{7}$

Step 1.4

Factor $3 x^{2}$ out of $- 18 x^{7}$ .

$3 x^{2} (3 x) + 3 x^{2} (- 2) + 3 x^{2} (4 x^{3}) + 3 x^{2} (- 6 x^{5})$

Step 1.5

Factor $3 x^{2}$ out of $3 x^{2} (3 x) + 3 x^{2} (- 2)$ .

$3 x^{2} (3 x - 2) + 3 x^{2} (4 x^{3}) + 3 x^{2} (- 6 x^{5})$

Step 1.6

Factor $3 x^{2}$ out of $3 x^{2} (3 x - 2) + 3 x^{2} (4 x^{3})$ .

$3 x^{2} (3 x - 2 + 4 x^{3}) + 3 x^{2} (- 6 x^{5})$

Step 1.7

Factor $3 x^{2}$ out of $3 x^{2} (3 x - 2 + 4 x^{3}) + 3 x^{2} (- 6 x^{5})$ .

$3 x^{2} (3 x - 2 + 4 x^{3} - 6 x^{5})$

Step 2

Regroup terms.

$3 x^{2} (- 6 x^{5} + 3 x - 2 + 4 x^{3})$

Step 3

Reorder terms.

$3 x^{2} (- 6 x^{5} + 4 x^{3} + 3 x - 2)$

Step 4

Factor $4 x^{3} + 3 x - 2$ using the rational roots test.

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Step 4.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 4, \pm 2$

Step 4.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

Step 4.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 4.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} + 3 \cdot 0.5 - 2$

Step 4.3.2

Raise $0.5$ to the power of $3$ .

$4 \cdot 0.125 + 3 \cdot 0.5 - 2$

Step 4.3.3

Multiply $4$ by $0.125$ .

$0.5 + 3 \cdot 0.5 - 2$

Step 4.3.4

Multiply $3$ by $0.5$ .

$0.5 + 1.5 - 2$

Step 4.3.5

Add $0.5$ and $1.5$ .

$2 - 2$

Step 4.3.6

Subtract $2$ from $2$ .

$0$

Step 4.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} + 3 x - 2}{2 x - 1}$

Step 4.5

Divide $4 x^{3} + 3 x - 2$ by $2 x - 1$ .

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Step 4.5.1

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

$2 x$

$1$

$4 x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 4.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$

Step 4.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			+	$4 x^{3}$	-	$2 x^{2}$

Step 4.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 2 x^{2}$

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$

Step 4.5.5

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$

Step 4.5.6

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

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 4.5.7

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

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 4.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$x$

Step 4.5.9

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

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$

Step 4.5.10

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

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$

Step 4.5.11

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

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$

Step 4.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$

Step 4.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							+	$4 x$	-	$2$

Step 4.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 2$

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							-	$4 x$	+	$2$

Step 4.5.15

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

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							-	$4 x$	+	$2$
										$0$

Step 4.5.16

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

$2 x^{2} + x + 2$

Step 4.6

Write $4 x^{3} + 3 x - 2$ as a set of factors.

$3 x^{2} (- 6 x^{5} + (2 x - 1) (2 x^{2} + x + 2))$

Step 5

Factor $- 1$ out of $- 6 x^{5} + (2 x - 1) (2 x^{2} + x + 2)$ .

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Step 5.1

Factor $- 1$ out of $- 6 x^{5}$ .

$3 x^{2} (- (6 x^{5}) + (2 x - 1) (2 x^{2} + x + 2))$

Step 5.2

Factor $- 1$ out of $(2 x - 1) (2 x^{2} + x + 2)$ .

$3 x^{2} (- (6 x^{5}) - ((- 2 x + 1) (2 x^{2} + x + 2)))$

Step 5.3

Factor $- 1$ out of $- (6 x^{5}) - ((- 2 x + 1) (2 x^{2} + x + 2))$ .

$3 x^{2} (- (6 x^{5} + (- 2 x + 1) (2 x^{2} + x + 2)))$

Step 6

Expand $(- 2 x + 1) (2 x^{2} + x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} (- (6 x^{5} - 2 x (2 x^{2}) - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7

Simplify each term.

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Step 7.1

Rewrite using the commutative property of multiplication.

$3 x^{2} (- (6 x^{5} - 2 \cdot 2 x \cdot x^{2} - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.2

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.2.1

Move $x^{2}$ .

$3 x^{2} (- (6 x^{5} - 2 \cdot 2 (x^{2} x) - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.2.2

Multiply $x^{2}$ by $x$ .

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Step 7.2.2.1

Raise $x$ to the power of $1$ .

$3 x^{2} (- (6 x^{5} - 2 \cdot 2 (x^{2} x^{1}) - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} (- (6 x^{5} - 2 \cdot 2 x^{2 + 1} - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.2.3

Add $2$ and $1$ .

$3 x^{2} (- (6 x^{5} - 2 \cdot 2 x^{3} - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.3

Multiply $- 2$ by $2$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x \cdot x - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.4

Multiply $x$ by $x$ by adding the exponents.

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Step 7.4.1

Move $x$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 (x \cdot x) - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.4.2

Multiply $x$ by $x$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x^{2} - 2 x \cdot 2 + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.5

Multiply $2$ by $- 2$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x^{2} - 4 x + 1 (2 x^{2}) + 1 x + 1 \cdot 2))$

Step 7.6

Multiply $2 x^{2}$ by $1$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x^{2} - 4 x + 2 x^{2} + 1 x + 1 \cdot 2))$

Step 7.7

Multiply $x$ by $1$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x^{2} - 4 x + 2 x^{2} + x + 1 \cdot 2))$

Step 7.8

Multiply $2$ by $1$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 2 x^{2} - 4 x + 2 x^{2} + x + 2))$

Step 8

Combine the opposite terms in $- 4 x^{3} - 2 x^{2} - 4 x + 2 x^{2} + x + 2$ .

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Step 8.1

Add $- 2 x^{2}$ and $2 x^{2}$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 4 x + 0 + x + 2))$

Step 8.2

Add $- 4 x^{3} - 4 x$ and $0$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 4 x + x + 2))$

Step 9

Add $- 4 x$ and $x$ .

$3 x^{2} (- (6 x^{5} - 4 x^{3} - 3 x + 2))$

Step 10

Remove unnecessary parentheses.

$3 x^{2} \cdot - 1 (6 x^{5} - 4 x^{3} - 3 x + 2)$

Step 11

Combine exponents.

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Step 11.1

Factor out negative.

$- (3 x^{2} (6 x^{5} - 4 x^{3} - 3 x + 2))$

Step 11.2

Multiply $3$ by $- 1$ .

$- 3 (x^{2} (6 x^{5} - 4 x^{3} - 3 x + 2))$

Step 12

Remove unnecessary parentheses.

$- 3 x^{2} (6 x^{5} - 4 x^{3} - 3 x + 2)$