Algebra Examples

$x^{13} - 2 x^{12} - x^{11} + 2 x^{10} = 0$

Step 1

Factor the left side of the equation.

Tap for more steps...

Step 1.1

Factor $x^{10}$ out of $x^{13} - 2 x^{12} - x^{11} + 2 x^{10}$ .

Tap for more steps...

Step 1.1.1

Factor $x^{10}$ out of $x^{13}$ .

$x^{10} x^{3} - 2 x^{12} - x^{11} + 2 x^{10} = 0$

Step 1.1.2

Factor $x^{10}$ out of $- 2 x^{12}$ .

$x^{10} x^{3} + x^{10} (- 2 x^{2}) - x^{11} + 2 x^{10} = 0$

Step 1.1.3

Factor $x^{10}$ out of $- x^{11}$ .

$x^{10} x^{3} + x^{10} (- 2 x^{2}) + x^{10} (- x) + 2 x^{10} = 0$

Step 1.1.4

Factor $x^{10}$ out of $2 x^{10}$ .

$x^{10} x^{3} + x^{10} (- 2 x^{2}) + x^{10} (- x) + x^{10} \cdot 2 = 0$

Step 1.1.5

Factor $x^{10}$ out of $x^{10} x^{3} + x^{10} (- 2 x^{2})$ .

$x^{10} (x^{3} - 2 x^{2}) + x^{10} (- x) + x^{10} \cdot 2 = 0$

Step 1.1.6

Factor $x^{10}$ out of $x^{10} (x^{3} - 2 x^{2}) + x^{10} (- x)$ .

$x^{10} (x^{3} - 2 x^{2} - x) + x^{10} \cdot 2 = 0$

Step 1.1.7

Factor $x^{10}$ out of $x^{10} (x^{3} - 2 x^{2} - x) + x^{10} \cdot 2$ .

$x^{10} (x^{3} - 2 x^{2} - x + 2) = 0$

Step 1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.2.1

Group the first two terms and the last two terms.

$x^{10} ((x^{3} - 2 x^{2}) - x + 2) = 0$

Step 1.2.2

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

$x^{10} (x^{2} (x - 2) - (x - 2)) = 0$

Step 1.3

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

$x^{10} ((x - 2) (x^{2} - 1)) = 0$

Step 1.4

Rewrite $1$ as $1^{2}$ .

$x^{10} ((x - 2) (x^{2} - 1^{2})) = 0$

Step 1.5

Factor.

Tap for more steps...

Step 1.5.1

Factor.

Tap for more steps...

Step 1.5.1.1

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 $b = 1$ .

$x^{10} ((x - 2) ((x + 1) (x - 1))) = 0$

Step 1.5.1.2

Remove unnecessary parentheses.

$x^{10} ((x - 2) (x + 1) (x - 1)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$x^{10} (x - 2) (x + 1) (x - 1) = 0$

Step 2

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

$x^{10} = 0$

$x - 2 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set $x^{10}$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 3.1

Set $x^{10}$ equal to $0$ .

$x^{10} = 0$

Step 3.2

Solve $x^{10} = 0$ for $x$ .

Tap for more steps...

Step 3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[10]{0}$

Step 3.2.2

Simplify $\pm \sqrt[10]{0}$ .

Tap for more steps...

Step 3.2.2.1

Rewrite $0$ as $0^{10}$ .

$x = \pm \sqrt[10]{0^{10}}$

Step 3.2.2.2

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

$x = \pm 0$

Step 3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4

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

Tap for more steps...

Step 4.1

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

$x - 2 = 0$

Step 4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5

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

Tap for more steps...

Step 5.1

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

$x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 6

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

Tap for more steps...

Step 6.1

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

$x - 1 = 0$

Step 6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 7

The final solution is all the values that make $x^{10} (x - 2) (x + 1) (x - 1) = 0$ true.

$x = 0, 2, - 1, 1$