Precalculus Examples

$9 x^{3} - 12 x^{2} - 8 x - 1 = 0$

Step 1

Factor $9 x^{3} - 12 x^{2} - 8 x - 1$ using the rational roots test.

Tap for more steps...

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.

$p = \pm 1$

$q = \pm 1, \pm 9, \pm 3$

Step 1.2

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

$\pm 1, \pm 0.\overline{1}, \pm 0.\overline{3}$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Substitute $- 0.\overline{3}$ into the polynomial.

$9 {(- 0.\overline{3})}^{3} - 12 {(- 0.\overline{3})}^{2} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.2

Raise $- 0.\overline{3}$ to the power of $3$ .

$9 \cdot - 0.\overline{037} - 12 {(- 0.\overline{3})}^{2} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.3

Multiply $9$ by $- 0.\overline{037}$ .

$- 0.\overline{3} - 12 {(- 0.\overline{3})}^{2} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.4

Raise $- 0.\overline{3}$ to the power of $2$ .

$- 0.\overline{3} - 12 \cdot 0.\overline{1} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.5

Multiply $- 12$ by $0.\overline{1}$ .

$- 0.\overline{3} - 1.\overline{3} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.6

Subtract $1.\overline{3}$ from $- 0.\overline{3}$ .

$- 1.\overline{6} - 8 \cdot - 0.\overline{3} - 1$

Step 1.3.7

Multiply $- 8$ by $- 0.\overline{3}$ .

$- 1.\overline{6} + 2.\overline{6} - 1$

Step 1.3.8

Add $- 1.\overline{6}$ and $2.\overline{6}$ .

$1 - 1$

Step 1.3.9

Subtract $1$ from $1$ .

$0$

Step 1.4

Since $- 0.\overline{3}$ is a known root, divide the polynomial by $3 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{9 x^{3} - 12 x^{2} - 8 x - 1}{3 x + 1}$

Step 1.5

Divide $9 x^{3} - 12 x^{2} - 8 x - 1$ by $3 x + 1$ .

Tap for more steps...

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 $0$ .

$3 x$

$1$

$9 x^{3}$

$12 x^{2}$

$8 x$

$1$

Step 1.5.2

Divide the highest order term in the dividend $9 x^{3}$ by the highest order term in divisor $3 x$ .

					$3 x^{2}$
	$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			+	$9 x^{3}$	+	$3 x^{2}$

Step 1.5.4

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$

Step 1.5.5

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$

Step 1.5.6

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

				$3 x^{2}$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$

Step 1.5.7

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $3 x$ .

				$3 x^{2}$	-	$5 x$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$5 x$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					-	$15 x^{2}$	-	$5 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} - 5 x$

				$3 x^{2}$	-	$5 x$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$

Step 1.5.10

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

				$3 x^{2}$	-	$5 x$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$

Step 1.5.11

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

				$3 x^{2}$	-	$5 x$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$	-	$1$

Step 1.5.12

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

				$3 x^{2}$	-	$5 x$	-	$1$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$	-	$1$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$5 x$	-	$1$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$	-	$1$
							-	$3 x$	-	$1$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x - 1$

				$3 x^{2}$	-	$5 x$	-	$1$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$	-	$1$
							+	$3 x$	+	$1$

Step 1.5.15

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

				$3 x^{2}$	-	$5 x$	-	$1$
$3 x$	+	$1$		$9 x^{3}$	-	$12 x^{2}$	-	$8 x$	-	$1$
			-	$9 x^{3}$	-	$3 x^{2}$
					-	$15 x^{2}$	-	$8 x$
					+	$15 x^{2}$	+	$5 x$
							-	$3 x$	-	$1$
							+	$3 x$	+	$1$
										$0$

Step 1.5.16

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

$3 x^{2} - 5 x - 1$

Step 1.6

Write $9 x^{3} - 12 x^{2} - 8 x - 1$ as a set of factors.

$(3 x + 1) (3 x^{2} - 5 x - 1) = 0$

Step 2

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

$3 x + 1 = 0$

$3 x^{2} - 5 x - 1 = 0$

Step 3

Set $3 x + 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 3.2

Solve $3 x + 1 = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 3.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

Tap for more steps...

Step 3.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 3.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 4

Set $3 x^{2} - 5 x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $3 x^{2} - 5 x - 1$ equal to $0$ .

$3 x^{2} - 5 x - 1 = 0$

Step 4.2

Solve $3 x^{2} - 5 x - 1 = 0$ for $x$ .

Tap for more steps...

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 $a = 3$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 4.2.3

Simplify.

Tap for more steps...

Step 4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

Tap for more steps...

Step 4.2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot - 1}}{2 \cdot 3}$

Step 4.2.3.1.2.2

Multiply $- 12$ by $- 1$ .

$x = \frac{5 \pm \sqrt{25 + 12}}{2 \cdot 3}$

Step 4.2.3.1.3

Add $25$ and $12$ .

$x = \frac{5 \pm \sqrt{37}}{2 \cdot 3}$

Step 4.2.3.2

Multiply $2$ by $3$ .

$x = \frac{5 \pm \sqrt{37}}{6}$

Step 4.2.4

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{37}}{6}, \frac{5 - \sqrt{37}}{6}$

Step 5

The final solution is all the values that make $(3 x + 1) (3 x^{2} - 5 x - 1) = 0$ true.

$x = - \frac{1}{3}, \frac{5 + \sqrt{37}}{6}, \frac{5 - \sqrt{37}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{3}, \frac{5 + \sqrt{37}}{6}, \frac{5 - \sqrt{37}}{6}$

Decimal Form:

$x = - 0.\overline{3}, 1.84712708 \dots, - 0.18046042 \dots$