Precalculus Examples

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

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x$ out of $x^{4} - 2 x^{3} - x^{2} + 2 x$ .

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Step 1.1.1

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

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

Step 1.1.2

Factor $x$ out of $- 2 x^{3}$ .

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

Step 1.1.3

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

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

Step 1.1.4

Factor $x$ out of $2 x$ .

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

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

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Factor.

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Step 1.5.1

Factor.

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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 $b = 1$ .

$x ((x - 2) ((x + 1) (x - 1))) = 0$

Step 1.5.1.2

Remove unnecessary parentheses.

$x ((x - 2) (x + 1) (x - 1)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$x (x - 2) (x + 1) (x - 1) = 0$

Step 2

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

$x = 0$

$x - 2 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

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

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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 $x$ .

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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 $x$ .

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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 $x (x - 2) (x + 1) (x - 1) = 0$ true.

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