Precalculus Examples

$f (x) = x^{5} - 3 x^{4} - 3 x^{3} + 9 x^{2} - 4 x + 12$

Step 1

Set $x^{5} - 3 x^{4} - 3 x^{3} + 9 x^{2} - 4 x + 12$ equal to $0$ .

$x^{5} - 3 x^{4} - 3 x^{3} + 9 x^{2} - 4 x + 12 = 0$

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.

$x^{5} - 3 x^{3} - 4 x - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.2

Factor $x$ out of $x^{5} - 3 x^{3} - 4 x$ .

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Step 2.1.2.1

Factor $x$ out of $x^{5}$ .

$x \cdot x^{4} - 3 x^{3} - 4 x - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.2.2

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

$x \cdot x^{4} + x (- 3 x^{2}) - 4 x - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.2.3

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

$x \cdot x^{4} + x (- 3 x^{2}) + x \cdot - 4 - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.2.4

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

$x (x^{4} - 3 x^{2}) + x \cdot - 4 - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.2.5

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

$x (x^{4} - 3 x^{2} - 4) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.3

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

$x ({(x^{2})}^{2} - 3 x^{2} - 4) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x (u^{2} - 3 u - 4) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.5

Factor $u^{2} - 3 u - 4$ using the AC method.

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Step 2.1.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 2.1.5.2

Write the factored form using these integers.

$x ((u - 4) (u + 1)) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.6

Replace all occurrences of $u$ with $x^{2}$ .

$x ((x^{2} - 4) (x^{2} + 1)) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.7

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

$x ((x^{2} - 2^{2}) (x^{2} + 1)) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.8

Factor.

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Step 2.1.8.1

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 $b = 2$ .

$x ((x + 2) (x - 2) (x^{2} + 1)) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.8.2

Remove unnecessary parentheses.

$x (x + 2) (x - 2) (x^{2} + 1) - 3 x^{4} + 9 x^{2} + 12 = 0$

Step 2.1.9

Factor $3$ out of $- 3 x^{4} + 9 x^{2} + 12$ .

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Step 2.1.9.1

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

$x (x + 2) (x - 2) (x^{2} + 1) + 3 (- x^{4}) + 9 x^{2} + 12 = 0$

Step 2.1.9.2

Factor $3$ out of $9 x^{2}$ .

$x (x + 2) (x - 2) (x^{2} + 1) + 3 (- x^{4}) + 3 (3 x^{2}) + 12 = 0$

Step 2.1.9.3

Factor $3$ out of $12$ .

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

Step 2.1.9.4

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

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

Step 2.1.9.5

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

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

Step 2.1.10

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

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

Step 2.1.11

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

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

Step 2.1.12

Factor by grouping.

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Step 2.1.12.1

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 $a \cdot c = - 1 \cdot 4 = - 4$ and whose sum is $b = 3$ .

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Step 2.1.12.1.1

Factor $3$ out of $3 u$ .

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

Step 2.1.12.1.2

Rewrite $3$ as $- 1$ plus $4$

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

Step 2.1.12.1.3

Apply the distributive property.

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

Step 2.1.12.2

Factor out the greatest common factor from each group.

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Step 2.1.12.2.1

Group the first two terms and the last two terms.

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

Step 2.1.12.2.2

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

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

Step 2.1.12.3

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

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

Step 2.1.13

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 2.1.14

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

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

Step 2.1.15

Factor.

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Step 2.1.15.1

Factor.

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Step 2.1.15.1.1

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 $b = 2$ .

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

Step 2.1.15.1.2

Remove unnecessary parentheses.

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

Step 2.1.15.2

Remove unnecessary parentheses.

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

Step 2.1.16

Factor $(x + 2) (x - 2)$ out of $x (x + 2) (x - 2) (x^{2} + 1) + 3 (- x^{2} - 1) (x + 2) (x - 2)$ .

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Step 2.1.16.1

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

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

Step 2.1.16.2

Factor $(x + 2) (x - 2)$ out of $3 (- x^{2} - 1) (x + 2) (x - 2)$ .

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

Step 2.1.16.3

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) ((x) (x^{2} + 1)) + (x + 2) (x - 2) (3 (- x^{2} - 1))$ .

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

Step 2.1.17

Apply the distributive property.

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

Step 2.1.18

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

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Step 2.1.18.1

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

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Step 2.1.18.1.1

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

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

Step 2.1.18.1.2

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

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

Step 2.1.18.2

Add $1$ and $2$ .

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

Step 2.1.19

Multiply $x$ by $1$ .

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

Step 2.1.20

Apply the distributive property.

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

Step 2.1.21

Multiply $- 1$ by $3$ .

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

Step 2.1.22

Multiply $3$ by $- 1$ .

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

Step 2.1.23

Reorder terms.

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

Step 2.1.24

Factor.

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Step 2.1.24.1

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

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Step 2.1.24.1.1

Factor out the greatest common factor from each group.

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Step 2.1.24.1.1.1

Group the first two terms and the last two terms.

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

Step 2.1.24.1.1.2

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

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

Step 2.1.24.1.2

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

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

Step 2.1.24.2

Remove unnecessary parentheses.

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

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 $0$ .

$x + 2 = 0$

$x - 2 = 0$

$x - 3 = 0$

$x^{2} + 1 = 0$

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

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Step 2.4.1

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

$x - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

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

$x - 3 = 0$

Step 2.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.6

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

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Step 2.6.1

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

$x^{2} + 1 = 0$

Step 2.6.2

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

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Step 2.6.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.6.2.2

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

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

Step 2.6.2.3

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

$x = \pm i$

Step 2.6.2.4

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

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Step 2.6.2.4.1

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

$x = i$

Step 2.6.2.4.2

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

$x = - i$

Step 2.6.2.4.3

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

$x = i, - i$

Step 2.7

The final solution is all the values that make $(x + 2) (x - 2) (x - 3) (x^{2} + 1) = 0$ true.

$x = - 2, 2, 3, i, - i$

Step 3