Finite Math Examples

$x^{4} + 4 x^{3} - 11 x^{2} - 26 x - 12 = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Factor $x^{4} + 4 x^{3} - 11 x^{2} - 26 x - 12$ using the rational roots test.

Tap for more steps...

Step 1.1.1

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 1.1.2

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 1.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

Tap for more steps...

Step 1.1.3.1

Substitute $- 1$ into the polynomial.

${(- 1)}^{4} + 4 {(- 1)}^{3} - 11 {(- 1)}^{2} - 26 \cdot - 1 - 12$

Step 1.1.3.2

Raise $- 1$ to the power of $4$ .

$1 + 4 {(- 1)}^{3} - 11 {(- 1)}^{2} - 26 \cdot - 1 - 12$

Step 1.1.3.3

Raise $- 1$ to the power of $3$ .

$1 + 4 \cdot - 1 - 11 {(- 1)}^{2} - 26 \cdot - 1 - 12$

Step 1.1.3.4

Multiply $4$ by $- 1$ .

$1 - 4 - 11 {(- 1)}^{2} - 26 \cdot - 1 - 12$

Step 1.1.3.5

Subtract $4$ from $1$ .

$- 3 - 11 {(- 1)}^{2} - 26 \cdot - 1 - 12$

Step 1.1.3.6

Raise $- 1$ to the power of $2$ .

$- 3 - 11 \cdot 1 - 26 \cdot - 1 - 12$

Step 1.1.3.7

Multiply $- 11$ by $1$ .

$- 3 - 11 - 26 \cdot - 1 - 12$

Step 1.1.3.8

Subtract $11$ from $- 3$ .

$- 14 - 26 \cdot - 1 - 12$

Step 1.1.3.9

Multiply $- 26$ by $- 1$ .

$- 14 + 26 - 12$

Step 1.1.3.10

Add $- 14$ and $26$ .

$12 - 12$

Step 1.1.3.11

Subtract $12$ from $12$ .

$0$

Step 1.1.4

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

$\frac{x^{4} + 4 x^{3} - 11 x^{2} - 26 x - 12}{x + 1}$

Step 1.1.5

Divide $x^{4} + 4 x^{3} - 11 x^{2} - 26 x - 12$ by $x + 1$ .

Tap for more steps...

Step 1.1.5.1

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

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

Step 1.1.5.2

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

Step 1.1.5.4

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

				$x^{3}$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$

Step 1.1.5.5

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

Step 1.1.5.6

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

$x^{3}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

Step 1.1.5.7

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 1.1.5.9

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

Step 1.1.5.10

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

Step 1.1.5.11

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

$x^{3}$

$3 x^{2}$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

$26 x$

Step 1.1.5.12

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $x$ .

$x^{3}$

$3 x^{2}$

$14 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

$26 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$3 x^{2}$	-	$14 x$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$11 x^{2}$
					-	$3 x^{3}$	-	$3 x^{2}$
							-	$14 x^{2}$	-	$26 x$
							-	$14 x^{2}$	-	$14 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} - 14 x$

$x^{3}$

$3 x^{2}$

$14 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

$26 x$

$14 x^{2}$

$14 x$

Step 1.1.5.15

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

$x^{3}$

$3 x^{2}$

$14 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

$26 x$

$14 x^{2}$

$14 x$

$12 x$

Step 1.1.5.16

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

$x^{3}$

$3 x^{2}$

$14 x$

$x$

$1$

$x^{4}$

$4 x^{3}$

$11 x^{2}$

$26 x$

$12$

$x^{4}$

$x^{3}$

$3 x^{3}$

$11 x^{2}$

$3 x^{3}$

$3 x^{2}$

$14 x^{2}$

$26 x$

$14 x^{2}$

$14 x$

$12 x$

$12$

Step 1.1.5.17

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

				$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$11 x^{2}$
					-	$3 x^{3}$	-	$3 x^{2}$
							-	$14 x^{2}$	-	$26 x$
							+	$14 x^{2}$	+	$14 x$
									-	$12 x$	-	$12$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$11 x^{2}$
					-	$3 x^{3}$	-	$3 x^{2}$
							-	$14 x^{2}$	-	$26 x$
							+	$14 x^{2}$	+	$14 x$
									-	$12 x$	-	$12$
									-	$12 x$	-	$12$

Step 1.1.5.19

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

				$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$11 x^{2}$
					-	$3 x^{3}$	-	$3 x^{2}$
							-	$14 x^{2}$	-	$26 x$
							+	$14 x^{2}$	+	$14 x$
									-	$12 x$	-	$12$
									+	$12 x$	+	$12$

Step 1.1.5.20

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

				$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
$x$	+	$1$		$x^{4}$	+	$4 x^{3}$	-	$11 x^{2}$	-	$26 x$	-	$12$
			-	$x^{4}$	-	$x^{3}$
					+	$3 x^{3}$	-	$11 x^{2}$
					-	$3 x^{3}$	-	$3 x^{2}$
							-	$14 x^{2}$	-	$26 x$
							+	$14 x^{2}$	+	$14 x$
									-	$12 x$	-	$12$
									+	$12 x$	+	$12$
												$0$

Step 1.1.5.21

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

$x^{3} + 3 x^{2} - 14 x - 12$

Step 1.1.6

Write $x^{4} + 4 x^{3} - 11 x^{2} - 26 x - 12$ as a set of factors.

$(x + 1) (x^{3} + 3 x^{2} - 14 x - 12) = 0$

Step 1.2

Factor $x^{3} + 3 x^{2} - 14 x - 12$ using the rational roots test.

Tap for more steps...

Step 1.2.1

Factor $x^{3} + 3 x^{2} - 14 x - 12$ using the rational roots test.

Tap for more steps...

Step 1.2.1.1

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.

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

$q = \pm 1$

Step 1.2.1.2

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 1.2.1.3

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

Tap for more steps...

Step 1.2.1.3.1

Substitute $3$ into the polynomial.

$3^{3} + 3 \cdot 3^{2} - 14 \cdot 3 - 12$

Step 1.2.1.3.2

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

$27 + 3 \cdot 3^{2} - 14 \cdot 3 - 12$

Step 1.2.1.3.3

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

$27 + 3 \cdot 9 - 14 \cdot 3 - 12$

Step 1.2.1.3.4

Multiply $3$ by $9$ .

$27 + 27 - 14 \cdot 3 - 12$

Step 1.2.1.3.5

Add $27$ and $27$ .

$54 - 14 \cdot 3 - 12$

Step 1.2.1.3.6

Multiply $- 14$ by $3$ .

$54 - 42 - 12$

Step 1.2.1.3.7

Subtract $42$ from $54$ .

$12 - 12$

Step 1.2.1.3.8

Subtract $12$ from $12$ .

$0$

Step 1.2.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 3 x^{2} - 14 x - 12}{x - 3}$

Step 1.2.1.5

Divide $x^{3} + 3 x^{2} - 14 x - 12$ by $x - 3$ .

Tap for more steps...

Step 1.2.1.5.1

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

$x$

$3$

$x^{3}$

$3 x^{2}$

$14 x$

$12$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			+	$x^{3}$	-	$3 x^{2}$

Step 1.2.1.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$

Step 1.2.1.5.7

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

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					+	$6 x^{2}$	-	$18 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 18 x$

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$

Step 1.2.1.5.10

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

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$

Step 1.2.1.5.11

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

				$x^{2}$	+	$6 x$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$	-	$12$

Step 1.2.1.5.12

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

				$x^{2}$	+	$6 x$	+	$4$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$	-	$12$

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$6 x$	+	$4$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$	-	$12$
							+	$4 x$	-	$12$

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 12$

				$x^{2}$	+	$6 x$	+	$4$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$	-	$12$
							-	$4 x$	+	$12$

Step 1.2.1.5.15

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

				$x^{2}$	+	$6 x$	+	$4$
$x$	-	$3$		$x^{3}$	+	$3 x^{2}$	-	$14 x$	-	$12$
			-	$x^{3}$	+	$3 x^{2}$
					+	$6 x^{2}$	-	$14 x$
					-	$6 x^{2}$	+	$18 x$
							+	$4 x$	-	$12$
							-	$4 x$	+	$12$
										$0$

Step 1.2.1.5.16

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

$x^{2} + 6 x + 4$

Step 1.2.1.6

Write $x^{3} + 3 x^{2} - 14 x - 12$ as a set of factors.

$(x + 1) ((x - 3) (x^{2} + 6 x + 4)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$(x + 1) (x - 3) (x^{2} + 6 x + 4) = 0$

Step 2

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

$x + 1 = 0$

$x - 3 = 0$

$x^{2} + 6 x + 4 = 0$

Step 3

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

Tap for more steps...

Step 3.1

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

$x + 1 = 0$

Step 3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

Set $x^{2} + 6 x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.1

Set $x^{2} + 6 x + 4$ equal to $0$ .

$x^{2} + 6 x + 4 = 0$

Step 5.2

Solve $x^{2} + 6 x + 4 = 0$ for $x$ .

Tap for more steps...

Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

Substitute the values $a = 1$ , $b = 6$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 5.2.3

Simplify.

Tap for more steps...

Step 5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

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

Tap for more steps...

Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Step 5.2.3.1.3

Subtract $16$ from $36$ .

$x = \frac{- 6 \pm \sqrt{20}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $20$ as $2^{2} \cdot 5$ .

Tap for more steps...

Step 5.2.3.1.4.1

Factor $4$ out of $20$ .

$x = \frac{- 6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

Step 5.2.3.1.4.2

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{5}}{2}$

Step 5.2.3.3

Simplify $\frac{- 6 \pm 2 \sqrt{5}}{2}$ .

$x = - 3 \pm \sqrt{5}$

Step 5.2.4

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{5}, - 3 - \sqrt{5}$

Step 6

The final solution is all the values that make $(x + 1) (x - 3) (x^{2} + 6 x + 4) = 0$ true.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 1, 3, - 0.76393202 \dots, - 5.23606797 \dots$