Calculus Examples

$x^{3} - x + 6$

Step 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

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 3

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

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Step 3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{3} - - 2 + 6$

Step 3.2

Raise $- 2$ to the power of $3$ .

$- 8 - - 2 + 6$

Step 3.3

Multiply $- 1$ by $- 2$ .

$- 8 + 2 + 6$

Step 3.4

Add $- 8$ and $2$ .

$- 6 + 6$

Step 3.5

Add $- 6$ and $6$ .

$0$

Step 4

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

$\frac{x^{3} - x + 6}{x + 2}$

Step 5

Divide $x^{3} - x + 6$ by $x + 2$ .

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Step 5.1

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

$x$

$2$

$x^{3}$

$0 x^{2}$

$x$

$6$

Step 5.2

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

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

Step 5.3

Multiply the new quotient term by the divisor.

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Multiply the new quotient term by the divisor.

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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

Step 5.12

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

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

Step 5.13

Multiply the new quotient term by the divisor.

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

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x + 6$

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

Step 5.15

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

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

Step 5.16

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

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

Step 6

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

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