Algebra Examples

$- 2 x^{3} + 14 x^{2} - 24 x + 20$

Step 1

Factor $2$ out of $- 2 x^{3} + 14 x^{2} - 24 x + 20$ .

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Step 1.1

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

$2 (- x^{3}) + 14 x^{2} - 24 x + 20$

Step 1.2

Factor $2$ out of $14 x^{2}$ .

$2 (- x^{3}) + 2 (7 x^{2}) - 24 x + 20$

Step 1.3

Factor $2$ out of $- 24 x$ .

$2 (- x^{3}) + 2 (7 x^{2}) + 2 (- 12 x) + 20$

Step 1.4

Factor $2$ out of $20$ .

$2 (- x^{3}) + 2 (7 x^{2}) + 2 (- 12 x) + 2 (10)$

Step 1.5

Factor $2$ out of $2 (- x^{3}) + 2 (7 x^{2})$ .

$2 (- x^{3} + 7 x^{2}) + 2 (- 12 x) + 2 (10)$

Step 1.6

Factor $2$ out of $2 (- x^{3} + 7 x^{2}) + 2 (- 12 x)$ .

$2 (- x^{3} + 7 x^{2} - 12 x) + 2 (10)$

Step 1.7

Factor $2$ out of $2 (- x^{3} + 7 x^{2} - 12 x) + 2 (10)$ .

$2 (- x^{3} + 7 x^{2} - 12 x + 10)$

Step 2

Factor.

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Step 2.1

Factor $- x^{3} + 7 x^{2} - 12 x + 10$ 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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 2.1.2

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 2.1.3

Substitute $5$ and simplify the expression. In this case, the expression is equal to $0$ so $5$ is a root of the polynomial.

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Step 2.1.3.1

Substitute $5$ into the polynomial.

$- 5^{3} + 7 \cdot 5^{2} - 12 \cdot 5 + 10$

Step 2.1.3.2

Raise $5$ to the power of $3$ .

$- 1 \cdot 125 + 7 \cdot 5^{2} - 12 \cdot 5 + 10$

Step 2.1.3.3

Multiply $- 1$ by $125$ .

$- 125 + 7 \cdot 5^{2} - 12 \cdot 5 + 10$

Step 2.1.3.4

Raise $5$ to the power of $2$ .

$- 125 + 7 \cdot 25 - 12 \cdot 5 + 10$

Step 2.1.3.5

Multiply $7$ by $25$ .

$- 125 + 175 - 12 \cdot 5 + 10$

Step 2.1.3.6

Add $- 125$ and $175$ .

$50 - 12 \cdot 5 + 10$

Step 2.1.3.7

Multiply $- 12$ by $5$ .

$50 - 60 + 10$

Step 2.1.3.8

Subtract $60$ from $50$ .

$- 10 + 10$

Step 2.1.3.9

Add $- 10$ and $10$ .

$0$

Step 2.1.4

Since $5$ is a known root, divide the polynomial by $x - 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{- x^{3} + 7 x^{2} - 12 x + 10}{x - 5}$

Step 2.1.5

Divide $- x^{3} + 7 x^{2} - 12 x + 10$ by $x - 5$ .

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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 $0$ .

$x$

$5$

$x^{3}$

$7 x^{2}$

$12 x$

$10$

Step 2.1.5.2

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

				-	$x^{2}$
	$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			-	$x^{3}$	+	$5 x^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{3} + 5 x^{2}$

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$

Step 2.1.5.5

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

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$

Step 2.1.5.6

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

			-	$x^{2}$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$

Step 2.1.5.7

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

			-	$x^{2}$	+	$2 x$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$2 x$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					+	$2 x^{2}$	-	$10 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 10 x$

			-	$x^{2}$	+	$2 x$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$

Step 2.1.5.10

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

			-	$x^{2}$	+	$2 x$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$

Step 2.1.5.11

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

			-	$x^{2}$	+	$2 x$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$	+	$10$

Step 2.1.5.12

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

			-	$x^{2}$	+	$2 x$	-	$2$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$	+	$10$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	+	$2 x$	-	$2$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$	+	$10$
							-	$2 x$	+	$10$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x + 10$

			-	$x^{2}$	+	$2 x$	-	$2$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$	+	$10$
							+	$2 x$	-	$10$

Step 2.1.5.15

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

			-	$x^{2}$	+	$2 x$	-	$2$
$x$	-	$5$	-	$x^{3}$	+	$7 x^{2}$	-	$12 x$	+	$10$
			+	$x^{3}$	-	$5 x^{2}$
					+	$2 x^{2}$	-	$12 x$
					-	$2 x^{2}$	+	$10 x$
							-	$2 x$	+	$10$
							+	$2 x$	-	$10$
										$0$

Step 2.1.5.16

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

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

Step 2.1.6

Write $- x^{3} + 7 x^{2} - 12 x + 10$ as a set of factors.

$2 ((x - 5) (- x^{2} + 2 x - 2))$

Step 2.2

Remove unnecessary parentheses.

$2 (x - 5) (- x^{2} + 2 x - 2)$

Step 3

Factor.

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Step 3.1

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

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Step 3.1.1

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

$2 (x - 5) (- (x^{2}) + 2 x - 2)$

Step 3.1.2

Factor $- 1$ out of $2 x$ .

$2 (x - 5) (- (x^{2}) - (- 2 x) - 2)$

Step 3.1.3

Rewrite $- 2$ as $- 1 (2)$ .

$2 (x - 5) (- (x^{2}) - (- 2 x) - 1 (2))$

Step 3.1.4

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

$2 (x - 5) (- (x^{2} - 2 x) - 1 (2))$

Step 3.1.5

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

$2 (x - 5) (- (x^{2} - 2 x + 2))$

Step 3.2

Remove unnecessary parentheses.

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

Step 4

Combine exponents.

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Step 4.1

Factor out negative.

$- (2 (x - 5) (x^{2} - 2 x + 2))$

Step 4.2

Multiply $2$ by $- 1$ .

$- 2 ((x - 5) (x^{2} - 2 x + 2))$

Step 5

Remove unnecessary parentheses.

$- 2 (x - 5) (x^{2} - 2 x + 2)$