Basic Math Examples

$(k^{3} - k^{2} - k - 2) \div (k - 2)$

Step 1

Factor $k^{3} - k^{2} - k - 2$ using the rational roots test.

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Step 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.

$p = \pm 1, \pm 2$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 2$

Step 1.3

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 1.3.1

Substitute $2$ into the polynomial.

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

Step 1.3.2

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

$8 - 2^{2} - 2 - 2$

Step 1.3.3

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

$8 - 1 \cdot 4 - 2 - 2$

Step 1.3.4

Multiply $- 1$ by $4$ .

$8 - 4 - 2 - 2$

Step 1.3.5

Subtract $4$ from $8$ .

$4 - 2 - 2$

Step 1.3.6

Subtract $2$ from $4$ .

$2 - 2$

Step 1.3.7

Subtract $2$ from $2$ .

$0$

Step 1.4

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

$\frac{k^{3} - k^{2} - k - 2}{k - 2}$

Step 1.5

Divide $k^{3} - k^{2} - k - 2$ by $k - 2$ .

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Step 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 $0$ .

$k$

$2$

$k^{3}$

$k^{2}$

$k$

$2$

Step 1.5.2

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

					$k^{2}$
	$k$	-	$2$		$k^{3}$	-	$k^{2}$	-	$k$	-	$2$

Step 1.5.3

Multiply the new quotient term by the divisor.

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

Step 1.5.4

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

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

Step 1.5.5

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

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

Step 1.5.6

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

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

Step 1.5.7

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

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

Step 1.5.8

Multiply the new quotient term by the divisor.

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

Step 1.5.9

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

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

Step 1.5.10

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

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

Step 1.5.11

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

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

Step 1.5.12

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

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

Step 1.5.13

Multiply the new quotient term by the divisor.

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

Step 1.5.14

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

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

Step 1.5.15

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

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

Step 1.5.16

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

$k^{2} + k + 1$

Step 1.6

Write $k^{3} - k^{2} - k - 2$ as a set of factors.

$(k - 2) (k^{2} + k + 1) \div (k - 2)$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Reduce the expression $(k - 2) (k^{2} + k + 1) \div (k - 2)$ by cancelling the common factors.

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Step 2.1.1

Cancel the common factor.

$(k - 2) (k^{2} + k + 1) \div (k - 2)$

Step 2.1.2

Rewrite the expression.

$\frac{k^{2} + k + 1}{1}$

Step 2.2

Divide $k^{2} + k + 1$ by $1$ .

$k^{2} + k + 1$