Precalculus Examples

$f (x) = x^{4} - 2 x^{3} - 6 x^{2} - 4 x - 16$

Step 1

Set $x^{4} - 2 x^{3} - 6 x^{2} - 4 x - 16$ equal to $0$ .

$x^{4} - 2 x^{3} - 6 x^{2} - 4 x - 16 = 0$

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

Regroup terms.

$x^{4} - 16 - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

${(x^{2})}^{2} - 16 - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.3

Rewrite $16$ as $4^{2}$ .

${(x^{2})}^{2} - 4^{2} - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 4$ .

$(x^{2} + 4) (x^{2} - 4) - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Rewrite $4$ as $2^{2}$ .

$(x^{2} + 4) (x^{2} - 2^{2}) - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.5.2

Factor.

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Step 2.1.5.2.1

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 $b = 2$ .

$(x^{2} + 4) ((x + 2) (x - 2)) - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.5.2.2

Remove unnecessary parentheses.

$(x^{2} + 4) (x + 2) (x - 2) - 2 x^{3} - 6 x^{2} - 4 x = 0$

Step 2.1.6

Factor $2 x$ out of $- 2 x^{3} - 6 x^{2} - 4 x$ .

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Step 2.1.6.1

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

$(x^{2} + 4) (x + 2) (x - 2) + 2 x (- x^{2}) - 6 x^{2} - 4 x = 0$

Step 2.1.6.2

Factor $2 x$ out of $- 6 x^{2}$ .

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

Step 2.1.6.3

Factor $2 x$ out of $- 4 x$ .

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

Step 2.1.6.4

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

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

Step 2.1.6.5

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

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

Step 2.1.7

Factor.

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Step 2.1.7.1

Factor by grouping.

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Step 2.1.7.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 2 = 2$ and whose sum is $b = - 3$ .

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Step 2.1.7.1.1.1

Factor $- 3$ out of $- 3 x$ .

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

Step 2.1.7.1.1.2

Rewrite $- 3$ as $- 1$ plus $- 2$

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

Step 2.1.7.1.1.3

Apply the distributive property.

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

Step 2.1.7.1.2

Factor out the greatest common factor from each group.

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Step 2.1.7.1.2.1

Group the first two terms and the last two terms.

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

Step 2.1.7.1.2.2

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

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

Step 2.1.7.1.3

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

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

Step 2.1.7.2

Remove unnecessary parentheses.

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

Step 2.1.8

Factor $x + 2$ out of $(x^{2} + 4) (x + 2) (x - 2) + 2 x (- x - 1) (x + 2)$ .

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Step 2.1.8.1

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

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

Step 2.1.8.2

Factor $x + 2$ out of $2 x (- x - 1) (x + 2)$ .

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

Step 2.1.8.3

Factor $x + 2$ out of $(x + 2) ((x^{2} + 4) (x - 2)) + (x + 2) (2 x (- x - 1))$ .

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

Step 2.1.9

Expand $(x^{2} + 4) (x - 2)$ using the FOIL Method.

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Step 2.1.9.1

Apply the distributive property.

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

Step 2.1.9.2

Apply the distributive property.

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

Step 2.1.9.3

Apply the distributive property.

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

Step 2.1.10

Simplify each term.

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Step 2.1.10.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.1.10.1.1

Multiply $x^{2}$ by $x$ .

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Step 2.1.10.1.1.1

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

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

Step 2.1.10.1.1.2

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

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

Step 2.1.10.1.2

Add $2$ and $1$ .

$(x + 2) (x^{3} + x^{2} \cdot - 2 + 4 x + 4 \cdot - 2 + 2 x (- x - 1)) = 0$

Step 2.1.10.2

Move $- 2$ to the left of $x^{2}$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x + 4 \cdot - 2 + 2 x (- x - 1)) = 0$

Step 2.1.10.3

Multiply $4$ by $- 2$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 x (- x - 1)) = 0$

Step 2.1.11

Apply the distributive property.

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 x (- x) + 2 x \cdot - 1) = 0$

Step 2.1.12

Rewrite using the commutative property of multiplication.

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 \cdot (- 1 x \cdot x) + 2 x \cdot - 1) = 0$

Step 2.1.13

Multiply $- 1$ by $2$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 \cdot (- 1 x \cdot x) - 2 x) = 0$

Step 2.1.14

Simplify each term.

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Step 2.1.14.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.1.14.1.1

Move $x$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 \cdot (- 1 (x \cdot x)) - 2 x) = 0$

Step 2.1.14.1.2

Multiply $x$ by $x$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 + 2 \cdot (- 1 x^{2}) - 2 x) = 0$

Step 2.1.14.2

Multiply $2$ by $- 1$ .

$(x + 2) (x^{3} - 2 x^{2} + 4 x - 8 - 2 x^{2} - 2 x) = 0$

Step 2.1.15

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$(x + 2) (x^{3} - 4 x^{2} + 4 x - 8 - 2 x) = 0$

Step 2.1.16

Subtract $2 x$ from $4 x$ .

$(x + 2) (x^{3} - 4 x^{2} + 2 x - 8) = 0$

Step 2.1.17

Factor.

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Step 2.1.17.1

Rewrite $x^{3} - 4 x^{2} + 2 x - 8$ in a factored form.

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Step 2.1.17.1.1

Factor out the greatest common factor from each group.

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Step 2.1.17.1.1.1

Group the first two terms and the last two terms.

$(x + 2) ((x^{3} - 4 x^{2}) + 2 x - 8) = 0$

Step 2.1.17.1.1.2

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

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

Step 2.1.17.1.2

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

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

Step 2.1.17.2

Remove unnecessary parentheses.

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

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 $0$ .

$x + 2 = 0$

$x - 4 = 0$

$x^{2} + 2 = 0$

Step 2.3

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

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Step 2.3.1

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

$x + 2 = 0$

Step 2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

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

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Step 2.5.1

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

$x^{2} + 2 = 0$

Step 2.5.2

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

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Step 2.5.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 2.5.2.2

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

$x = \pm \sqrt{- 2}$

Step 2.5.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 2.5.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 2.5.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Step 2.5.2.3.3

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

$x = \pm i \sqrt{2}$

Step 2.5.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.5.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i \sqrt{2}$

Step 2.5.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i \sqrt{2}$

Step 2.5.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i \sqrt{2}, - i \sqrt{2}$

Step 2.6

The final solution is all the values that make $(x + 2) (x - 4) (x^{2} + 2) = 0$ true.

$x = - 2, 4, i \sqrt{2}, - i \sqrt{2}$

Step 3