Algebra Examples

$x^{3} + 3 x + 2 x + 6$

Step 1

Add $3 x$ and $2 x$ .

$x^{3} + 5 x + 6$

Step 2

Factor $x^{3} + 5 x + 6$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 2.3

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 2.3.1

Substitute $- 1$ into the polynomial.

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

Step 2.3.2

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

$- 1 + 5 \cdot - 1 + 6$

Step 2.3.3

Multiply $5$ by $- 1$ .

$- 1 - 5 + 6$

Step 2.3.4

Subtract $5$ from $- 1$ .

$- 6 + 6$

Step 2.3.5

Add $- 6$ and $6$ .

$0$

Step 2.4

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.

$\frac{x^{3} + 5 x + 6}{x + 1}$

Step 2.5

Divide $x^{3} + 5 x + 6$ by $x + 1$ .

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Step 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 $0$ .

$x$

$1$

$x^{3}$

$0 x^{2}$

$5 x$

$6$

Step 2.5.2

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

					$x^{2}$
	$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			+	$x^{3}$	+	$x^{2}$

Step 2.5.4

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$

Step 2.5.5

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$

Step 2.5.6

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

				$x^{2}$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$

Step 2.5.7

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					-	$x^{2}$	-	$x$

Step 2.5.9

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$

Step 2.5.10

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$

Step 2.5.11

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

				$x^{2}$	-	$x$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$	+	$6$

Step 2.5.12

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$x$	+	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$	+	$6$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$x$	+	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$	+	$6$
							+	$6 x$	+	$6$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 6$

				$x^{2}$	-	$x$	+	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$

Step 2.5.15

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

				$x^{2}$	-	$x$	+	$6$
$x$	+	$1$		$x^{3}$	+	$0 x^{2}$	+	$5 x$	+	$6$
			-	$x^{3}$	-	$x^{2}$
					-	$x^{2}$	+	$5 x$
					+	$x^{2}$	+	$x$
							+	$6 x$	+	$6$
							-	$6 x$	-	$6$
										$0$

Step 2.5.16

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

$x^{2} - x + 6$

Step 2.6

Write $x^{3} + 5 x + 6$ as a set of factors.

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