Precalculus Examples

$2 x^{3} - 23 x^{2} + 78 x - 80 = 0$

Step 1

Factor $2 x^{3} - 23 x^{2} + 78 x - 80$ 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.

$p = \pm 1, \pm 80, \pm 2, \pm 40, \pm 4, \pm 20, \pm 5, \pm 16, \pm 8, \pm 10$

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 80, \pm 40, \pm 2, \pm 20, \pm 4, \pm 10, \pm 5, \pm 2.5, \pm 16, \pm 8$

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.

$2 \cdot 2^{3} - 23 \cdot 2^{2} + 78 \cdot 2 - 80$

Step 1.3.2

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

$2 \cdot 8 - 23 \cdot 2^{2} + 78 \cdot 2 - 80$

Step 1.3.3

Multiply $2$ by $8$ .

$16 - 23 \cdot 2^{2} + 78 \cdot 2 - 80$

Step 1.3.4

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

$16 - 23 \cdot 4 + 78 \cdot 2 - 80$

Step 1.3.5

Multiply $- 23$ by $4$ .

$16 - 92 + 78 \cdot 2 - 80$

Step 1.3.6

Subtract $92$ from $16$ .

$- 76 + 78 \cdot 2 - 80$

Step 1.3.7

Multiply $78$ by $2$ .

$- 76 + 156 - 80$

Step 1.3.8

Add $- 76$ and $156$ .

$80 - 80$

Step 1.3.9

Subtract $80$ from $80$ .

$0$

Step 1.4

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

$\frac{2 x^{3} - 23 x^{2} + 78 x - 80}{x - 2}$

Step 1.5

Divide $2 x^{3} - 23 x^{2} + 78 x - 80$ by $x - 2$ .

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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 $0$ .

$x$

$2$

$2 x^{3}$

$23 x^{2}$

$78 x$

$80$

Step 1.5.2

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

					$2 x^{2}$
	$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 1.5.4

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 1.5.5

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$

Step 1.5.6

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$

Step 1.5.7

Divide the highest order term in the dividend $- 19 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$19 x$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$19 x$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					-	$19 x^{2}$	+	$38 x$

Step 1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 19 x^{2} + 38 x$

				$2 x^{2}$	-	$19 x$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$

Step 1.5.10

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

				$2 x^{2}$	-	$19 x$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$

Step 1.5.11

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

				$2 x^{2}$	-	$19 x$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$	-	$80$

Step 1.5.12

Divide the highest order term in the dividend $40 x$ by the highest order term in divisor $x$ .

				$2 x^{2}$	-	$19 x$	+	$40$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$	-	$80$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$19 x$	+	$40$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$	-	$80$
							+	$40 x$	-	$80$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $40 x - 80$

				$2 x^{2}$	-	$19 x$	+	$40$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$	-	$80$
							-	$40 x$	+	$80$

Step 1.5.15

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

				$2 x^{2}$	-	$19 x$	+	$40$
$x$	-	$2$		$2 x^{3}$	-	$23 x^{2}$	+	$78 x$	-	$80$
			-	$2 x^{3}$	+	$4 x^{2}$
					-	$19 x^{2}$	+	$78 x$
					+	$19 x^{2}$	-	$38 x$
							+	$40 x$	-	$80$
							-	$40 x$	+	$80$
										$0$

Step 1.5.16

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

$2 x^{2} - 19 x + 40$

Step 1.6

Write $2 x^{3} - 23 x^{2} + 78 x - 80$ as a set of factors.

$(x - 2) (2 x^{2} - 19 x + 40) = 0$

Step 2

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

$x - 2 = 0$

$2 x^{2} - 19 x + 40 = 0$

Step 3

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4

Set $2 x^{2} - 19 x + 40$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $2 x^{2} - 19 x + 40$ equal to $0$ .

$2 x^{2} - 19 x + 40 = 0$

Step 4.2

Solve $2 x^{2} - 19 x + 40 = 0$ 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 $a = 2$ , $b = - 19$ , and $c = 40$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (2 \cdot 40)}}{2 \cdot 2}$

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 $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 40}}{2 \cdot 2}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 40$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 40}}{2 \cdot 2}$

Step 4.2.3.1.2.2

Multiply $- 8$ by $40$ .

$x = \frac{19 \pm \sqrt{361 - 320}}{2 \cdot 2}$

Step 4.2.3.1.3

Subtract $320$ from $361$ .

$x = \frac{19 \pm \sqrt{41}}{2 \cdot 2}$

Step 4.2.3.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm \sqrt{41}}{4}$

Step 4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 40}}{2 \cdot 2}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 2 \cdot 40$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 40}}{2 \cdot 2}$

Step 4.2.4.1.2.2

Multiply $- 8$ by $40$ .

$x = \frac{19 \pm \sqrt{361 - 320}}{2 \cdot 2}$

Step 4.2.4.1.3

Subtract $320$ from $361$ .

$x = \frac{19 \pm \sqrt{41}}{2 \cdot 2}$

Step 4.2.4.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm \sqrt{41}}{4}$

Step 4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{19 + \sqrt{41}}{4}$

Step 4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 40}}{2 \cdot 2}$

Step 4.2.5.1.2

Multiply $- 4 \cdot 2 \cdot 40$ .

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Step 4.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 40}}{2 \cdot 2}$

Step 4.2.5.1.2.2

Multiply $- 8$ by $40$ .

$x = \frac{19 \pm \sqrt{361 - 320}}{2 \cdot 2}$

Step 4.2.5.1.3

Subtract $320$ from $361$ .

$x = \frac{19 \pm \sqrt{41}}{2 \cdot 2}$

Step 4.2.5.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm \sqrt{41}}{4}$

Step 4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{19 - \sqrt{41}}{4}$

Step 4.2.6

The final answer is the combination of both solutions.

$x = \frac{19 + \sqrt{41}}{4}, \frac{19 - \sqrt{41}}{4}$

Step 5

The final solution is all the values that make $(x - 2) (2 x^{2} - 19 x + 40) = 0$ true.

$x = 2, \frac{19 + \sqrt{41}}{4}, \frac{19 - \sqrt{41}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 2, \frac{19 + \sqrt{41}}{4}, \frac{19 - \sqrt{41}}{4}$

Decimal Form:

$x = 2, 6.35078105 \dots, 3.14921894 \dots$