Solve the Inequality for x x^3+2x^2-4x-8>0

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

Step 1

Convert the inequality to an equation.

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

Step 2

Factor the left side of the equation.

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

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

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

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

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$(x + 2) ((x + 2) (x - 2)) = 0$

Remove unnecessary parentheses.

$(x + 2) (x + 2) (x - 2) = 0$

Combine exponents.

Raise $x + 2$ to the power of $1$ .

$(x + 2) (x + 2) (x - 2) = 0$

Raise $x + 2$ to the power of $1$ .

$(x + 2) (x + 2) (x - 2) = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Step 3

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

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

$x - 2 = 0$

Step 4

Set ${(x + 2)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x + 2)}^{2}$ equal to $0$ .

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

Solve ${(x + 2)}^{2} = 0$ for $x$ .

Set the $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 6

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

$x = - 2, 2$

Step 7

The solution consists of all of the true intervals.

$x > 2$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x > 2$

Interval Notation:

$(2, \infty)$

Step 9

Solve the Inequality for x x^3+2x^2-4x-8>0

Related Questions

Support LCMGCF.com by disabling your adblocker.