Pre-Algebra Examples

$x^{3} - 34 x - 12 = 0$

Step 1

Factor $x^{3} - 34 x - 12$ 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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 1.3

Substitute $6$ and simplify the expression. In this case, the expression is equal to $0$ so $6$ is a root of the polynomial.

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Step 1.3.1

Substitute $6$ into the polynomial.

$6^{3} - 34 \cdot 6 - 12$

Step 1.3.2

Raise $6$ to the power of $3$ .

$216 - 34 \cdot 6 - 12$

Step 1.3.3

Multiply $- 34$ by $6$ .

$216 - 204 - 12$

Step 1.3.4

Subtract $204$ from $216$ .

$12 - 12$

Step 1.3.5

Subtract $12$ from $12$ .

$0$

Step 1.4

Since $6$ is a known root, divide the polynomial by $x - 6$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 34 x - 12}{x - 6}$

Step 1.5

Divide $x^{3} - 34 x - 12$ by $x - 6$ .

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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 $0$ .

$x$

$6$

$x^{3}$

$0 x^{2}$

$34 x$

$12$

Step 1.5.2

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

					$x^{2}$
	$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			+	$x^{3}$	-	$6 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 6 x^{2}$

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$

Step 1.5.5

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$

Step 1.5.6

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

				$x^{2}$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$

Step 1.5.7

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					+	$6 x^{2}$	-	$36 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 36 x$

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$

Step 1.5.10

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

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$

Step 1.5.11

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

				$x^{2}$	+	$6 x$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$	-	$12$

Step 1.5.12

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

				$x^{2}$	+	$6 x$	+	$2$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$	-	$12$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$2$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$	-	$12$
							+	$2 x$	-	$12$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 x - 12$

				$x^{2}$	+	$6 x$	+	$2$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$	-	$12$
							-	$2 x$	+	$12$

Step 1.5.15

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

				$x^{2}$	+	$6 x$	+	$2$
$x$	-	$6$		$x^{3}$	+	$0 x^{2}$	-	$34 x$	-	$12$
			-	$x^{3}$	+	$6 x^{2}$
					+	$6 x^{2}$	-	$34 x$
					-	$6 x^{2}$	+	$36 x$
							+	$2 x$	-	$12$
							-	$2 x$	+	$12$
										$0$

Step 1.5.16

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

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

Step 1.6

Write $x^{3} - 34 x - 12$ as a set of factors.

$(x - 6) (x^{2} + 6 x + 2) = 0$

Step 2

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

$x - 6 = 0$

$x^{2} + 6 x + 2 = 0$

Step 3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 4

Set $x^{2} + 6 x + 2$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 6 x + 2$ equal to $0$ .

$x^{2} + 6 x + 2 = 0$

Step 4.2

Solve $x^{2} + 6 x + 2 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $8$ from $36$ .

$x = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.2.3.1.4.1

Factor $4$ out of $28$ .

$x = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

Step 4.2.3.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Step 4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Step 4.2.3.3

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$x = - 3 \pm \sqrt{7}$

Step 4.2.4

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 5

The final solution is all the values that make $(x - 6) (x^{2} + 6 x + 2) = 0$ true.

$x = 6, - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 6, - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Decimal Form:

$x = 6, - 0.35424868 \dots, - 5.64575131 \dots$