Basic Math Examples

$w^{3} - w^{2} - 2 w = 40$

Step 1

Subtract $40$ from both sides of the equation.

$w^{3} - w^{2} - 2 w - 40 = 0$

Step 2

Factor $w^{3} - w^{2} - 2 w - 40$ 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.

$p = \pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

$q = \pm 1$

Step 2.2

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

$\pm 1, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 8$

Step 2.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

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Step 2.3.1

Substitute $4$ into the polynomial.

$4^{3} - 4^{2} - 2 \cdot 4 - 40$

Step 2.3.2

Raise $4$ to the power of $3$ .

$64 - 4^{2} - 2 \cdot 4 - 40$

Step 2.3.3

Raise $4$ to the power of $2$ .

$64 - 1 \cdot 16 - 2 \cdot 4 - 40$

Step 2.3.4

Multiply $- 1$ by $16$ .

$64 - 16 - 2 \cdot 4 - 40$

Step 2.3.5

Subtract $16$ from $64$ .

$48 - 2 \cdot 4 - 40$

Step 2.3.6

Multiply $- 2$ by $4$ .

$48 - 8 - 40$

Step 2.3.7

Subtract $8$ from $48$ .

$40 - 40$

Step 2.3.8

Subtract $40$ from $40$ .

$0$

Step 2.4

Since $4$ is a known root, divide the polynomial by $w - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{w^{3} - w^{2} - 2 w - 40}{w - 4}$

Step 2.5

Divide $w^{3} - w^{2} - 2 w - 40$ by $w - 4$ .

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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 $0$ .

$w$

$4$

$w^{3}$

$w^{2}$

$2 w$

$40$

Step 2.5.2

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

					$w^{2}$
	$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$

Step 2.5.3

Multiply the new quotient term by the divisor.

				$w^{2}$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			+	$w^{3}$	-	$4 w^{2}$

Step 2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $w^{3} - 4 w^{2}$

				$w^{2}$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$

Step 2.5.5

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

				$w^{2}$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$

Step 2.5.6

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

				$w^{2}$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$

Step 2.5.7

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

				$w^{2}$	+	$3 w$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$

Step 2.5.8

Multiply the new quotient term by the divisor.

				$w^{2}$	+	$3 w$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					+	$3 w^{2}$	-	$12 w$

Step 2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 w^{2} - 12 w$

				$w^{2}$	+	$3 w$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$

Step 2.5.10

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

				$w^{2}$	+	$3 w$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$

Step 2.5.11

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

				$w^{2}$	+	$3 w$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$	-	$40$

Step 2.5.12

Divide the highest order term in the dividend $10 w$ by the highest order term in divisor $w$ .

				$w^{2}$	+	$3 w$	+	$10$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$	-	$40$

Step 2.5.13

Multiply the new quotient term by the divisor.

				$w^{2}$	+	$3 w$	+	$10$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$	-	$40$
							+	$10 w$	-	$40$

Step 2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 w - 40$

				$w^{2}$	+	$3 w$	+	$10$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$	-	$40$
							-	$10 w$	+	$40$

Step 2.5.15

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

				$w^{2}$	+	$3 w$	+	$10$
$w$	-	$4$		$w^{3}$	-	$w^{2}$	-	$2 w$	-	$40$
			-	$w^{3}$	+	$4 w^{2}$
					+	$3 w^{2}$	-	$2 w$
					-	$3 w^{2}$	+	$12 w$
							+	$10 w$	-	$40$
							-	$10 w$	+	$40$
										$0$

Step 2.5.16

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

$w^{2} + 3 w + 10$

Step 2.6

Write $w^{3} - w^{2} - 2 w - 40$ as a set of factors.

$(w - 4) (w^{2} + 3 w + 10) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$w - 4 = 0$

$w^{2} + 3 w + 10 = 0$

Step 4

Set $w - 4$ equal to $0$ and solve for $w$ .

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Step 4.1

Set $w - 4$ equal to $0$ .

$w - 4 = 0$

Step 4.2

Add $4$ to both sides of the equation.

$w = 4$

Step 5

Set $w^{2} + 3 w + 10$ equal to $0$ and solve for $w$ .

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Step 5.1

Set $w^{2} + 3 w + 10$ equal to $0$ .

$w^{2} + 3 w + 10 = 0$

Step 5.2

Solve $w^{2} + 3 w + 10 = 0$ for $w$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 10$ into the quadratic formula and solve for $w$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

$w = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$w = \frac{- 3 \pm \sqrt{9 - 4 \cdot 10}}{2 \cdot 1}$

Step 5.2.3.1.2.2

Multiply $- 4$ by $10$ .

$w = \frac{- 3 \pm \sqrt{9 - 40}}{2 \cdot 1}$

Step 5.2.3.1.3

Subtract $40$ from $9$ .

$w = \frac{- 3 \pm \sqrt{- 31}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$w = \frac{- 3 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 1}$

Step 5.2.3.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$w = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 1}$

Step 5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$w = \frac{- 3 \pm i \sqrt{31}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$w = \frac{- 3 \pm i \sqrt{31}}{2}$

Step 5.2.4

The final answer is the combination of both solutions.

$w = - \frac{3 - i \sqrt{31}}{2}, - \frac{3 + i \sqrt{31}}{2}$

Step 6

The final solution is all the values that make $(w - 4) (w^{2} + 3 w + 10) = 0$ true.

$w = 4, - \frac{3 - i \sqrt{31}}{2}, - \frac{3 + i \sqrt{31}}{2}$