Finite Math Examples

$8 x^{3} - 10 x^{2} - 7 x - 1 = 0$

Step 1

Factor $8 x^{3} - 10 x^{2} - 7 x - 1$ 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.

$p = \pm 1$

$q = \pm 1, \pm 8, \pm 2, \pm 4$

Step 1.2

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

$\pm 1, \pm 0.125, \pm 0.5, \pm 0.25$

Step 1.3

Substitute $- 0.25$ and simplify the expression. In this case, the expression is equal to $0$ so $- 0.25$ is a root of the polynomial.

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Step 1.3.1

Substitute $- 0.25$ into the polynomial.

$8 {(- 0.25)}^{3} - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 1.3.2

Raise $- 0.25$ to the power of $3$ .

$8 \cdot - 0.015625 - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 1.3.3

Multiply $8$ by $- 0.015625$ .

$- 0.125 - 10 {(- 0.25)}^{2} - 7 \cdot - 0.25 - 1$

Step 1.3.4

Raise $- 0.25$ to the power of $2$ .

$- 0.125 - 10 \cdot 0.0625 - 7 \cdot - 0.25 - 1$

Step 1.3.5

Multiply $- 10$ by $0.0625$ .

$- 0.125 - 0.625 - 7 \cdot - 0.25 - 1$

Step 1.3.6

Subtract $0.625$ from $- 0.125$ .

$- 0.75 - 7 \cdot - 0.25 - 1$

Step 1.3.7

Multiply $- 7$ by $- 0.25$ .

$- 0.75 + 1.75 - 1$

Step 1.3.8

Add $- 0.75$ and $1.75$ .

$1 - 1$

Step 1.3.9

Subtract $1$ from $1$ .

$0$

Step 1.4

Since $- 0.25$ is a known root, divide the polynomial by $4 x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{8 x^{3} - 10 x^{2} - 7 x - 1}{4 x + 1}$

Step 1.5

Divide $8 x^{3} - 10 x^{2} - 7 x - 1$ by $4 x + 1$ .

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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 $0$ .

$4 x$

$1$

$8 x^{3}$

$10 x^{2}$

$7 x$

$1$

Step 1.5.2

Divide the highest order term in the dividend $8 x^{3}$ by the highest order term in divisor $4 x$ .

					$2 x^{2}$
	$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			+	$8 x^{3}$	+	$2 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{3} + 2 x^{2}$

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$

Step 1.5.5

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

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$

Step 1.5.6

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

				$2 x^{2}$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$

Step 1.5.7

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					-	$12 x^{2}$	-	$3 x$

Step 1.5.9

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

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$

Step 1.5.10

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

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$

Step 1.5.11

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

				$2 x^{2}$	-	$3 x$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$

Step 1.5.12

Divide the highest order term in the dividend $- 4 x$ by the highest order term in divisor $4 x$ .

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							-	$4 x$	-	$1$

Step 1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x - 1$

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							+	$4 x$	+	$1$

Step 1.5.15

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

				$2 x^{2}$	-	$3 x$	-	$1$
$4 x$	+	$1$		$8 x^{3}$	-	$10 x^{2}$	-	$7 x$	-	$1$
			-	$8 x^{3}$	-	$2 x^{2}$
					-	$12 x^{2}$	-	$7 x$
					+	$12 x^{2}$	+	$3 x$
							-	$4 x$	-	$1$
							+	$4 x$	+	$1$
										$0$

Step 1.5.16

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

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

Step 1.6

Write $8 x^{3} - 10 x^{2} - 7 x - 1$ as a set of factors.

$(4 x + 1) (2 x^{2} - 3 x - 1) = 0$

Step 2

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

$4 x + 1 = 0$

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

Step 3

Set $4 x + 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $4 x + 1$ equal to $0$ .

$4 x + 1 = 0$

Step 3.2

Solve $4 x + 1 = 0$ for $x$ .

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$4 x = - 1$

Step 3.2.2

Divide each term in $4 x = - 1$ by $4$ and simplify.

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Step 3.2.2.1

Divide each term in $4 x = - 1$ by $4$ .

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $4$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 1}{4}$

Step 3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{4}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 4

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

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Step 4.1

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

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

Step 4.2

Solve $2 x^{2} - 3 x - 1 = 0$ 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 $a = 2$ , $b = - 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

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

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

Step 4.2.3.1.2

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

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Step 4.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 4.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 4.2.3.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 4.2.3.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

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

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

Step 4.2.4.1.2

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

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Step 4.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 4.2.4.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 4.2.4.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 4.2.4.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{17}}{4}$

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

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

Step 4.2.5.1.2

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

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Step 4.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 4.2.5.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 4.2.5.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 4.2.5.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{17}}{4}$

Step 4.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 5

The final solution is all the values that make $(4 x + 1) (2 x^{2} - 3 x - 1) = 0$ true.

$x = - \frac{1}{4}, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{4}, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Decimal Form:

$x = - 0.25, 1.78077640 \dots, - 0.28077640 \dots$