Precalculus Examples

$x^{4} - 3 x^{3} - 27 x + 81$

Step 1

Set $x^{4} - 3 x^{3} - 27 x + 81$ equal to $0$ .

$x^{4} - 3 x^{3} - 27 x + 81 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor out the greatest common factor from each group.

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Step 2.1.1.1

Group the first two terms and the last two terms.

$(x^{4} - 3 x^{3}) - 27 x + 81 = 0$

Step 2.1.1.2

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

$x^{3} (x - 3) - 27 (x - 3) = 0$

Step 2.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 3$ .

$(x - 3) (x^{3} - 27) = 0$

Step 2.1.3

Rewrite $27$ as $3^{3}$ .

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

Step 2.1.4

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

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

Step 2.1.5

Factor.

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Step 2.1.5.1

Simplify.

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Step 2.1.5.1.1

Move $3$ to the left of $x$ .

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

Step 2.1.5.1.2

Raise $3$ to the power of $2$ .

$(x - 3) ((x - 3) (x^{2} + 3 x + 9)) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$(x - 3) (x - 3) (x^{2} + 3 x + 9) = 0$

Step 2.1.6

Combine exponents.

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Step 2.1.6.1

Raise $x - 3$ to the power of $1$ .

$(x - 3) (x - 3) (x^{2} + 3 x + 9) = 0$

Step 2.1.6.2

Raise $x - 3$ to the power of $1$ .

$(x - 3) (x - 3) (x^{2} + 3 x + 9) = 0$

Step 2.1.6.3

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

${(x - 3)}^{1 + 1} (x^{2} + 3 x + 9) = 0$

Step 2.1.6.4

Add $1$ and $1$ .

${(x - 3)}^{2} (x^{2} + 3 x + 9) = 0$

Step 2.2

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

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

$x^{2} + 3 x + 9 = 0$

Step 2.3

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 2.3.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 2.3.2.1

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.3.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.4

Set $x^{2} + 3 x + 9$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} + 3 x + 9$ equal to $0$ .

$x^{2} + 3 x + 9 = 0$

Step 2.4.2

Solve $x^{2} + 3 x + 9 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.2.3.1.6

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

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 2.4.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.4.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 2.4.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

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

Step 2.4.2.3.1.8

Pull terms out from under the radical.

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

Step 2.4.2.3.1.9

Move $3$ to the left of $i$ .

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

Step 2.4.2.3.2

Multiply $2$ by $1$ .

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

Step 2.4.2.4

The final answer is the combination of both solutions.

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

Step 2.5

The final solution is all the values that make ${(x - 3)}^{2} (x^{2} + 3 x + 9) = 0$ true.

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

Step 3