Precalculus Examples

$x^{4} - x^{3} + x - 1 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$(x^{4} - x^{3}) + x - 1 = 0$

Step 1.1.2

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

$x^{3} (x - 1) + 1 (x - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

$(x - 1) (x^{3} + 1) = 0$

Step 1.3

Rewrite $1$ as $1^{3}$ .

$(x - 1) (x^{3} + 1^{3}) = 0$

Step 1.4

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 1.5

Factor.

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Step 1.5.1

Simplify.

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Step 1.5.1.1

Multiply $- 1$ by $1$ .

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

Step 1.5.1.2

One to any power is one.

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

Step 1.5.2

Remove unnecessary parentheses.

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

Step 2

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

$x - 1 = 0$

$x + 1 = 0$

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

Step 3

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

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Step 3.1

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

$x - 1 = 0$

Step 3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4

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

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x^{2} - x + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - x + 1$ equal to $0$ .

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

Step 5.2

Solve $x^{2} - x + 1 = 0$ for $x$ .

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Step 5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Raise $- 1$ to the power of $2$ .

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

Step 5.2.3.1.2

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

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 5.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 5.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 5.2.4

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 6

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

$x = 1, - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$