Algebra Examples

$x^{4} - 4 x^{3} + 8 x \geq 0$

Step 1

Convert the inequality to an equation.

$x^{4} - 4 x^{3} + 8 x = 0$

Step 2

Factor the left side of the equation.

Tap for more steps...

Step 2.1

Factor $x$ out of $x^{4} - 4 x^{3} + 8 x$ .

Tap for more steps...

Step 2.1.1

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} - 4 x^{3} + 8 x = 0$

Step 2.1.2

Factor $x$ out of $- 4 x^{3}$ .

$x \cdot x^{3} + x (- 4 x^{2}) + 8 x = 0$

Step 2.1.3

Factor $x$ out of $8 x$ .

$x \cdot x^{3} + x (- 4 x^{2}) + x \cdot 8 = 0$

Step 2.1.4

Factor $x$ out of $x \cdot x^{3} + x (- 4 x^{2})$ .

$x (x^{3} - 4 x^{2}) + x \cdot 8 = 0$

Step 2.1.5

Factor $x$ out of $x (x^{3} - 4 x^{2}) + x \cdot 8$ .

$x (x^{3} - 4 x^{2} + 8) = 0$

Step 2.2

Factor.

Tap for more steps...

Step 2.2.1

Factor $x^{3} - 4 x^{2} + 8$ using the rational roots test.

Tap for more steps...

Step 2.2.1.1

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.

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

$q = \pm 1$

Step 2.2.1.2

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

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

Step 2.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

Tap for more steps...

Step 2.2.1.3.1

Substitute $2$ into the polynomial.

$2^{3} - 4 \cdot 2^{2} + 8$

Step 2.2.1.3.2

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

$8 - 4 \cdot 2^{2} + 8$

Step 2.2.1.3.3

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

$8 - 4 \cdot 4 + 8$

Step 2.2.1.3.4

Multiply $- 4$ by $4$ .

$8 - 16 + 8$

Step 2.2.1.3.5

Subtract $16$ from $8$ .

$- 8 + 8$

Step 2.2.1.3.6

Add $- 8$ and $8$ .

$0$

Step 2.2.1.4

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.

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

Step 2.2.1.5

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

Tap for more steps...

Step 2.2.1.5.1

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

$x$

$2$

$x^{3}$

$4 x^{2}$

$0 x$

$8$

Step 2.2.1.5.2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$

Step 2.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.2.1.5.4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.2.1.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$

Step 2.2.1.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$

Step 2.2.1.5.7

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$

Step 2.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					-	$2 x^{2}$	+	$4 x$

Step 2.2.1.5.9

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$

Step 2.2.1.5.10

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$

Step 2.2.1.5.11

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

				$x^{2}$	-	$2 x$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$	+	$8$

Step 2.2.1.5.12

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

				$x^{2}$	-	$2 x$	-	$4$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$	+	$8$

Step 2.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$2 x$	-	$4$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$	+	$8$
							-	$4 x$	+	$8$

Step 2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x + 8$

				$x^{2}$	-	$2 x$	-	$4$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$	+	$8$
							+	$4 x$	-	$8$

Step 2.2.1.5.15

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

				$x^{2}$	-	$2 x$	-	$4$
$x$	-	$2$		$x^{3}$	-	$4 x^{2}$	+	$0 x$	+	$8$
			-	$x^{3}$	+	$2 x^{2}$
					-	$2 x^{2}$	+	$0 x$
					+	$2 x^{2}$	-	$4 x$
							-	$4 x$	+	$8$
							+	$4 x$	-	$8$
										$0$

Step 2.2.1.5.16

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

$x^{2} - 2 x - 4$

Step 2.2.1.6

Write $x^{3} - 4 x^{2} + 8$ as a set of factors.

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

Step 2.2.2

Remove unnecessary parentheses.

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

Step 3

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

$x = 0$

$x - 2 = 0$

$x^{2} - 2 x - 4 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

Tap for more steps...

Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

Tap for more steps...

Step 6.1

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

$x^{2} - 2 x - 4 = 0$

Step 6.2

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

Tap for more steps...

Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 6.2.3

Simplify.

Tap for more steps...

Step 6.2.3.1

Simplify the numerator.

Tap for more steps...

Step 6.2.3.1.1

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

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

Step 6.2.3.1.2

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

Tap for more steps...

Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.2.3.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 6.2.3.1.3

Add $4$ and $16$ .

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

Step 6.2.3.1.4

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

Tap for more steps...

Step 6.2.3.1.4.1

Factor $4$ out of $20$ .

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

Step 6.2.3.1.4.2

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

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

Step 6.2.3.1.5

Pull terms out from under the radical.

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

Step 6.2.3.2

Multiply $2$ by $1$ .

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

Step 6.2.3.3

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

$x = 1 \pm \sqrt{5}$

Step 6.2.4

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

Tap for more steps...

Step 6.2.4.1

Simplify the numerator.

Tap for more steps...

Step 6.2.4.1.1

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

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

Step 6.2.4.1.2

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

Tap for more steps...

Step 6.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.2.4.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 6.2.4.1.3

Add $4$ and $16$ .

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

Step 6.2.4.1.4

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

Tap for more steps...

Step 6.2.4.1.4.1

Factor $4$ out of $20$ .

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

Step 6.2.4.1.4.2

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

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

Step 6.2.4.1.5

Pull terms out from under the radical.

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

Step 6.2.4.2

Multiply $2$ by $1$ .

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

Step 6.2.4.3

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

$x = 1 \pm \sqrt{5}$

Step 6.2.4.4

Change the $\pm$ to $+$ .

$x = 1 + \sqrt{5}$

Step 6.2.5

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

Tap for more steps...

Step 6.2.5.1

Simplify the numerator.

Tap for more steps...

Step 6.2.5.1.1

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

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

Step 6.2.5.1.2

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

Tap for more steps...

Step 6.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 6.2.5.1.2.2

Multiply $- 4$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 16}}{2 \cdot 1}$

Step 6.2.5.1.3

Add $4$ and $16$ .

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

Step 6.2.5.1.4

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

Tap for more steps...

Step 6.2.5.1.4.1

Factor $4$ out of $20$ .

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

Step 6.2.5.1.4.2

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

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

Step 6.2.5.1.5

Pull terms out from under the radical.

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

Step 6.2.5.2

Multiply $2$ by $1$ .

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

Step 6.2.5.3

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

$x = 1 \pm \sqrt{5}$

Step 6.2.5.4

Change the $\pm$ to $-$ .

$x = 1 - \sqrt{5}$

Step 6.2.6

The final answer is the combination of both solutions.

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

Step 7

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

$x = 0, 2, 1 + \sqrt{5}, 1 - \sqrt{5}$

Step 8

Use each root to create test intervals.

$x < 1 - \sqrt{5}$

$1 - \sqrt{5} < x < 0$

$0 < x < 2$

$2 < x < 1 + \sqrt{5}$

$x > 1 + \sqrt{5}$

Step 9

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 9.1

Test a value on the interval $x < 1 - \sqrt{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 9.1.1

Choose a value on the interval $x < 1 - \sqrt{5}$ and see if this value makes the original inequality true.

$x = - 4$

Step 9.1.2

Replace $x$ with $- 4$ in the original inequality.

${(- 4)}^{4} - 4 {(- 4)}^{3} + 8 (- 4) \geq 0$

Step 9.1.3

The left side $480$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.2

Test a value on the interval $1 - \sqrt{5} < x < 0$ to see if it makes the inequality true.

Tap for more steps...

Step 9.2.1

Choose a value on the interval $1 - \sqrt{5} < x < 0$ and see if this value makes the original inequality true.

$x = - 1$

Step 9.2.2

Replace $x$ with $- 1$ in the original inequality.

${(- 1)}^{4} - 4 {(- 1)}^{3} + 8 (- 1) \geq 0$

Step 9.2.3

The left side $- 3$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.3

Test a value on the interval $0 < x < 2$ to see if it makes the inequality true.

Tap for more steps...

Step 9.3.1

Choose a value on the interval $0 < x < 2$ and see if this value makes the original inequality true.

$x = 1$

Step 9.3.2

Replace $x$ with $1$ in the original inequality.

${(1)}^{4} - 4 {(1)}^{3} + 8 (1) \geq 0$

Step 9.3.3

The left side $5$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.4

Test a value on the interval $2 < x < 1 + \sqrt{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 9.4.1

Choose a value on the interval $2 < x < 1 + \sqrt{5}$ and see if this value makes the original inequality true.

$x = 3$

Step 9.4.2

Replace $x$ with $3$ in the original inequality.

${(3)}^{4} - 4 {(3)}^{3} + 8 (3) \geq 0$

Step 9.4.3

The left side $- 3$ is less than the right side $0$ , which means that the given statement is false.

False

Step 9.5

Test a value on the interval $x > 1 + \sqrt{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 9.5.1

Choose a value on the interval $x > 1 + \sqrt{5}$ and see if this value makes the original inequality true.

$x = 6$

Step 9.5.2

Replace $x$ with $6$ in the original inequality.

${(6)}^{4} - 4 {(6)}^{3} + 8 (6) \geq 0$

Step 9.5.3

The left side $480$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 9.6

Compare the intervals to determine which ones satisfy the original inequality.

$x < 1 - \sqrt{5}$ True

$1 - \sqrt{5} < x < 0$ False

$0 < x < 2$ True

$2 < x < 1 + \sqrt{5}$ False

$x > 1 + \sqrt{5}$ True

$x < 1 - \sqrt{5}$ True

$1 - \sqrt{5} < x < 0$ False

$0 < x < 2$ True

$2 < x < 1 + \sqrt{5}$ False

$x > 1 + \sqrt{5}$ True

Step 10

The solution consists of all of the true intervals.

$x \leq 1 - \sqrt{5}$ or $0 \leq x \leq 2$ or $x \geq 1 + \sqrt{5}$

Step 11

Convert the inequality to interval notation.

$(- \infty, 1 - \sqrt{5}] \cup [0, 2] \cup [1 + \sqrt{5}, \infty)$

Step 12