Basic Math Examples

$y^{3} - 11 y - 6$

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 $- 3$ and simplify the expression. In this case, the expression is equal to $0$ so $- 3$ is a root of the polynomial.

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Step 3.1

Substitute $- 3$ into the polynomial.

${(- 3)}^{3} - 11 \cdot - 3 - 6$

Step 3.2

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

$- 27 - 11 \cdot - 3 - 6$

Step 3.3

Multiply $- 11$ by $- 3$ .

$- 27 + 33 - 6$

Step 3.4

Add $- 27$ and $33$ .

$6 - 6$

Step 3.5

Subtract $6$ from $6$ .

$0$

Step 4

Since $- 3$ is a known root, divide the polynomial by $y + 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{y^{3} - 11 y - 6}{y + 3}$

Step 5

Divide $y^{3} - 11 y - 6$ by $y + 3$ .

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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 $0$ .

$y$

$3$

$y^{3}$

$0 y^{2}$

$11 y$

$6$

Step 5.2

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

					$y^{2}$
	$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$

Step 5.3

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			+	$y^{3}$	+	$3 y^{2}$

Step 5.4

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

				$y^{2}$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$

Step 5.5

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

				$y^{2}$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$

Step 5.6

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

				$y^{2}$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$

Step 5.7

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

				$y^{2}$	-	$3 y$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$

Step 5.8

Multiply the new quotient term by the divisor.

				$y^{2}$	-	$3 y$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					-	$3 y^{2}$	-	$9 y$

Step 5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 y^{2} - 9 y$

				$y^{2}$	-	$3 y$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$

Step 5.10

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

				$y^{2}$	-	$3 y$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$

Step 5.11

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

				$y^{2}$	-	$3 y$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$	-	$6$

Step 5.12

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

				$y^{2}$	-	$3 y$	-	$2$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$	-	$6$

Step 5.13

Multiply the new quotient term by the divisor.

				$y^{2}$	-	$3 y$	-	$2$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$	-	$6$
							-	$2 y$	-	$6$

Step 5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 y - 6$

				$y^{2}$	-	$3 y$	-	$2$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$	-	$6$
							+	$2 y$	+	$6$

Step 5.15

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

				$y^{2}$	-	$3 y$	-	$2$
$y$	+	$3$		$y^{3}$	+	$0 y^{2}$	-	$11 y$	-	$6$
			-	$y^{3}$	-	$3 y^{2}$
					-	$3 y^{2}$	-	$11 y$
					+	$3 y^{2}$	+	$9 y$
							-	$2 y$	-	$6$
							+	$2 y$	+	$6$
										$0$

Step 5.16

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

$y^{2} - 3 y - 2$

Step 6

Write $y^{3} - 11 y - 6$ as a set of factors.

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