Algebra Examples

$(x^{3} + 2 x - 1) - (2 x^{2} + 4 x - 2)$

Step 1

Apply the distributive property.

$x^{3} + 2 x - 1 - (2 x^{2}) - (4 x) - - 2$

Step 2

Simplify.

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Step 2.1

Multiply $2$ by $- 1$ .

$x^{3} + 2 x - 1 - 2 x^{2} - (4 x) - - 2$

Step 2.2

Multiply $4$ by $- 1$ .

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

Step 2.3

Multiply $- 1$ by $- 2$ .

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

Step 3

Subtract $4 x$ from $2 x$ .

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

Step 4

Add $- 1$ and $2$ .

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

Step 5

Reorder terms.

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

Step 6

Factor $x^{3} - 2 x^{2} - 2 x + 1$ using the rational roots test.

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Step 6.1

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.

$p = \pm 1$

$q = \pm 1$

Step 6.2

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

$\pm 1$

Step 6.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 6.3.1

Substitute $- 1$ into the polynomial.

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

Step 6.3.2

Raise $- 1$ to the power of $3$ .

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

Step 6.3.3

Raise $- 1$ to the power of $2$ .

$- 1 - 2 \cdot 1 - 2 \cdot - 1 + 1$

Step 6.3.4

Multiply $- 2$ by $1$ .

$- 1 - 2 - 2 \cdot - 1 + 1$

Step 6.3.5

Subtract $2$ from $- 1$ .

$- 3 - 2 \cdot - 1 + 1$

Step 6.3.6

Multiply $- 2$ by $- 1$ .

$- 3 + 2 + 1$

Step 6.3.7

Add $- 3$ and $2$ .

$- 1 + 1$

Step 6.3.8

Add $- 1$ and $1$ .

$0$

Step 6.4

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

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

Step 6.5

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

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Step 6.5.1

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

$x$

$1$

$x^{3}$

$2 x^{2}$

$2 x$

$1$

Step 6.5.2

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

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

Step 6.5.3

Multiply the new quotient term by the divisor.

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

Step 6.5.4

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

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

Step 6.5.5

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

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

Step 6.5.6

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

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

Step 6.5.7

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

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

Step 6.5.8

Multiply the new quotient term by the divisor.

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

Step 6.5.9

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

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

Step 6.5.10

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

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

Step 6.5.11

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

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

Step 6.5.12

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

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

Step 6.5.13

Multiply the new quotient term by the divisor.

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

Step 6.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x + 1$

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

Step 6.5.15

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

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

Step 6.5.16

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

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

Step 6.6

Write $x^{3} - 2 x^{2} - 2 x + 1$ as a set of factors.

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