Precalculus Examples

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

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

Step 1

Factor the left side of the equation.

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Step 1.1

Factor

x

$x$ out of

x^{4} - 4 x^{3} - x^{2} + 4 x

$x^{4} - 4 x^{3} - x^{2} + 4 x$ .

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Step 1.1.1

Factor

x

$x$ out of

x^{4}

$x^{4}$ .

x \cdot x^{3} - 4 x^{3} - x^{2} + 4 x = 0

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

Step 1.1.2

Factor

x

$x$ out of

- 4 x^{3}

$- 4 x^{3}$ .

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

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

Step 1.1.3

Factor

x

$x$ out of

- x^{2}

$- x^{2}$ .

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

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

Step 1.1.4

Factor

$x$ out of

$4 x$ .

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

Step 1.1.5

Factor

$x$ out of

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

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

Step 1.1.6

Factor

$x$ out of

$x (x^{3} - 4 x^{2}) + x (- x)$ .

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

Step 1.1.7

Factor

$x$ out of

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

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

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.

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

Step 1.2.2

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

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

Step 1.3

Factor the polynomial by factoring out the greatest common factor,

$x - 4$ .

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

Step 1.4

Rewrite

$1$ as

$1^{2}$ .

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

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

$b = 1$ .

$x ((x - 4) ((x + 1) (x - 1))) = 0$

Step 1.5.1.2

Remove unnecessary parentheses.

$x ((x - 4) (x + 1) (x - 1)) = 0$

Step 1.5.2

Remove unnecessary parentheses.

$x (x - 4) (x + 1) (x - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to

$0$ , the entire expression will be equal to

$0$ .

$x = 0$

$x - 4 = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 3

Set

$x$ equal to

$0$ .

$x = 0$

Step 4

Set

$x - 4$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$x - 4$ equal to

$0$ .

$x - 4 = 0$

Step 4.2

Add

$4$ to both sides of the equation.

$x = 4$

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

$x (x - 4) (x + 1) (x - 1) = 0$ true.

$x = 0, 4, - 1, 1$