Finite Math Examples

$5 x^{3} + 27 x^{2} + 11 x + 5$

Step 1

Set $5 x^{3} + 27 x^{2} + 11 x + 5$ equal to $0$ .

$5 x^{3} + 27 x^{2} + 11 x + 5 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Factor $5 x^{3} + 27 x^{2} + 11 x + 5$ using the rational roots test.

Tap for more steps...

Step 2.1.1

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.

$p = \pm 1, \pm 5$

$q = \pm 1, \pm 5$

Step 2.1.2

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

$\pm 1, \pm 0.2, \pm 5$

Step 2.1.3

Substitute $- 5$ and simplify the expression. In this case, the expression is equal to $0$ so $- 5$ is a root of the polynomial.

Tap for more steps...

Step 2.1.3.1

Substitute $- 5$ into the polynomial.

$5 {(- 5)}^{3} + 27 {(- 5)}^{2} + 11 \cdot - 5 + 5$

Step 2.1.3.2

Raise $- 5$ to the power of $3$ .

$5 \cdot - 125 + 27 {(- 5)}^{2} + 11 \cdot - 5 + 5$

Step 2.1.3.3

Multiply $5$ by $- 125$ .

$- 625 + 27 {(- 5)}^{2} + 11 \cdot - 5 + 5$

Step 2.1.3.4

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

$- 625 + 27 \cdot 25 + 11 \cdot - 5 + 5$

Step 2.1.3.5

Multiply $27$ by $25$ .

$- 625 + 675 + 11 \cdot - 5 + 5$

Step 2.1.3.6

Add $- 625$ and $675$ .

$50 + 11 \cdot - 5 + 5$

Step 2.1.3.7

Multiply $11$ by $- 5$ .

$50 - 55 + 5$

Step 2.1.3.8

Subtract $55$ from $50$ .

$- 5 + 5$

Step 2.1.3.9

Add $- 5$ and $5$ .

$0$

Step 2.1.4

Since $- 5$ is a known root, divide the polynomial by $x + 5$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{5 x^{3} + 27 x^{2} + 11 x + 5}{x + 5}$

Step 2.1.5

Divide $5 x^{3} + 27 x^{2} + 11 x + 5$ by $x + 5$ .

Tap for more steps...

Step 2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$5$

$5 x^{3}$

$27 x^{2}$

$11 x$

$5$

Step 2.1.5.2

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

					$5 x^{2}$
	$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$

Step 2.1.5.3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			+	$5 x^{3}$	+	$25 x^{2}$

Step 2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 25 x^{2}$

				$5 x^{2}$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$

Step 2.1.5.5

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

				$5 x^{2}$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$

Step 2.1.5.6

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

				$5 x^{2}$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$

Step 2.1.5.7

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

				$5 x^{2}$	+	$2 x$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$

Step 2.1.5.8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$2 x$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					+	$2 x^{2}$	+	$10 x$

Step 2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 10 x$

				$5 x^{2}$	+	$2 x$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$

Step 2.1.5.10

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

				$5 x^{2}$	+	$2 x$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$

Step 2.1.5.11

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

				$5 x^{2}$	+	$2 x$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$	+	$5$

Step 2.1.5.12

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

				$5 x^{2}$	+	$2 x$	+	$1$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$	+	$5$

Step 2.1.5.13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$2 x$	+	$1$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$	+	$5$
							+	$x$	+	$5$

Step 2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $x + 5$

				$5 x^{2}$	+	$2 x$	+	$1$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$	+	$5$
							-	$x$	-	$5$

Step 2.1.5.15

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

				$5 x^{2}$	+	$2 x$	+	$1$
$x$	+	$5$		$5 x^{3}$	+	$27 x^{2}$	+	$11 x$	+	$5$
			-	$5 x^{3}$	-	$25 x^{2}$
					+	$2 x^{2}$	+	$11 x$
					-	$2 x^{2}$	-	$10 x$
							+	$x$	+	$5$
							-	$x$	-	$5$
										$0$

Step 2.1.5.16

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

$5 x^{2} + 2 x + 1$

Step 2.1.6

Write $5 x^{3} + 27 x^{2} + 11 x + 5$ as a set of factors.

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

Step 2.2

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

$x + 5 = 0$

$5 x^{2} + 2 x + 1 = 0$

Step 2.3

Set $x + 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.3.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 2.3.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$5 x^{2} + 2 x + 1 = 0$

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 5$ , $b = 2$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 2.4.2.3

Simplify.

Tap for more steps...

Step 2.4.2.3.1

Simplify the numerator.

Tap for more steps...

Step 2.4.2.3.1.1

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

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

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 5 \cdot 1$ .

Tap for more steps...

Step 2.4.2.3.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20 \cdot 1}}{2 \cdot 5}$

Step 2.4.2.3.1.2.2

Multiply $- 20$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 20}}{2 \cdot 5}$

Step 2.4.2.3.1.3

Subtract $20$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 16}}{2 \cdot 5}$

Step 2.4.2.3.1.4

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 16}}{2 \cdot 5}$

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot 5}$

Step 2.4.2.3.1.6

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

$x = \frac{- 2 \pm i \cdot \sqrt{16}}{2 \cdot 5}$

Step 2.4.2.3.1.7

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4^{2}}}{2 \cdot 5}$

Step 2.4.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 \pm i \cdot 4}{2 \cdot 5}$

Step 2.4.2.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{- 2 \pm 4 i}{2 \cdot 5}$

Step 2.4.2.3.2

Multiply $2$ by $5$ .

$x = \frac{- 2 \pm 4 i}{10}$

Step 2.4.2.3.3

Simplify $\frac{- 2 \pm 4 i}{10}$ .

$x = \frac{- 1 \pm 2 i}{5}$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = - \frac{1}{5} + \frac{2 i}{5}, - \frac{1}{5} - \frac{2 i}{5}$

Step 2.5

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

$x = - 5, - \frac{1}{5} + \frac{2 i}{5}, - \frac{1}{5} - \frac{2 i}{5}$

Step 3