Precalculus Examples

$p (x) = 3 x^{4} + 7 x^{3} - 10 x^{2} - 28 x - 8$

Step 1

Set $3 x^{4} + 7 x^{3} - 10 x^{2} - 28 x - 8$ equal to $0$ .

$3 x^{4} + 7 x^{3} - 10 x^{2} - 28 x - 8 = 0$

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.

$7 x^{3} - 28 x + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.2

Factor $7 x$ out of $7 x^{3} - 28 x$ .

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Step 2.1.2.1

Factor $7 x$ out of $7 x^{3}$ .

$7 x (x^{2}) - 28 x + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.2.2

Factor $7 x$ out of $- 28 x$ .

$7 x (x^{2}) + 7 x (- 4) + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.2.3

Factor $7 x$ out of $7 x (x^{2}) + 7 x (- 4)$ .

$7 x (x^{2} - 4) + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.3

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

$7 x (x^{2} - 2^{2}) + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

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 $b = 2$ .

$7 x ((x + 2) (x - 2)) + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$7 x (x + 2) (x - 2) + 3 x^{4} - 10 x^{2} - 8 = 0$

Step 2.1.5

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

$7 x (x + 2) (x - 2) + 3 {(x^{2})}^{2} - 10 x^{2} - 8 = 0$

Step 2.1.6

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

$7 x (x + 2) (x - 2) + 3 u^{2} - 10 u - 8 = 0$

Step 2.1.7

Factor by grouping.

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Step 2.1.7.1

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 $a \cdot c = 3 \cdot - 8 = - 24$ and whose sum is $b = - 10$ .

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Step 2.1.7.1.1

Factor $- 10$ out of $- 10 u$ .

$7 x (x + 2) (x - 2) + 3 u^{2} - 10 u - 8 = 0$

Step 2.1.7.1.2

Rewrite $- 10$ as $2$ plus $- 12$

$7 x (x + 2) (x - 2) + 3 u^{2} + (2 - 12) u - 8 = 0$

Step 2.1.7.1.3

Apply the distributive property.

$7 x (x + 2) (x - 2) + 3 u^{2} + 2 u - 12 u - 8 = 0$

Step 2.1.7.2

Factor out the greatest common factor from each group.

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Step 2.1.7.2.1

Group the first two terms and the last two terms.

$7 x (x + 2) (x - 2) + 3 u^{2} + 2 u - 12 u - 8 = 0$

Step 2.1.7.2.2

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

$7 x (x + 2) (x - 2) + u (3 u + 2) - 4 (3 u + 2) = 0$

Step 2.1.7.3

Factor the polynomial by factoring out the greatest common factor, $3 u + 2$ .

$7 x (x + 2) (x - 2) + (3 u + 2) (u - 4) = 0$

Step 2.1.8

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

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x^{2} - 4) = 0$

Step 2.1.9

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

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x^{2} - 2^{2}) = 0$

Step 2.1.10

Factor.

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Step 2.1.10.1

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 $b = 2$ .

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) ((x + 2) (x - 2)) = 0$

Step 2.1.10.2

Remove unnecessary parentheses.

$7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x + 2) (x - 2) = 0$

Step 2.1.11

Factor $(x + 2) (x - 2)$ out of $7 x (x + 2) (x - 2) + (3 x^{2} + 2) (x + 2) (x - 2)$ .

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Step 2.1.11.1

Factor $(x + 2) (x - 2)$ out of $7 x (x + 2) (x - 2)$ .

$(x + 2) (x - 2) (7 x) + (3 x^{2} + 2) (x + 2) (x - 2) = 0$

Step 2.1.11.2

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

$(x + 2) (x - 2) (7 x) + (x + 2) (x - 2) (3 x^{2} + 2) = 0$

Step 2.1.11.3

Factor $(x + 2) (x - 2)$ out of $(x + 2) (x - 2) (7 x) + (x + 2) (x - 2) (3 x^{2} + 2)$ .

$(x + 2) (x - 2) (7 x + 3 x^{2} + 2) = 0$

Step 2.1.12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 2) (x - 2) (7 u + 3 u^{2} + 2) = 0$

Step 2.1.13

Factor by grouping.

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Step 2.1.13.1

Reorder terms.

$(x + 2) (x - 2) (3 u^{2} + 7 u + 2) = 0$

Step 2.1.13.2

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 $a \cdot c = 3 \cdot 2 = 6$ and whose sum is $b = 7$ .

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Step 2.1.13.2.1

Factor $7$ out of $7 u$ .

$(x + 2) (x - 2) (3 u^{2} + 7 (u) + 2) = 0$

Step 2.1.13.2.2

Rewrite $7$ as $1$ plus $6$

$(x + 2) (x - 2) (3 u^{2} + (1 + 6) u + 2) = 0$

Step 2.1.13.2.3

Apply the distributive property.

$(x + 2) (x - 2) (3 u^{2} + 1 u + 6 u + 2) = 0$

Step 2.1.13.2.4

Multiply $u$ by $1$ .

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

Step 2.1.13.3

Factor out the greatest common factor from each group.

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Step 2.1.13.3.1

Group the first two terms and the last two terms.

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

Step 2.1.13.3.2

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

$(x + 2) (x - 2) (u (3 u + 1) + 2 (3 u + 1)) = 0$

Step 2.1.13.4

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$(x + 2) (x - 2) ((3 u + 1) (u + 2)) = 0$

Step 2.1.14

Factor.

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Step 2.1.14.1

Replace all occurrences of $u$ with $x$ .

$(x + 2) (x - 2) ((3 x + 1) (x + 2)) = 0$

Step 2.1.14.2

Remove unnecessary parentheses.

$(x + 2) (x - 2) (3 x + 1) (x + 2) = 0$

Step 2.1.15

Combine exponents.

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Step 2.1.15.1

Raise $x + 2$ to the power of $1$ .

$(x + 2) (x + 2) (x - 2) (3 x + 1) = 0$

Step 2.1.15.2

Raise $x + 2$ to the power of $1$ .

$(x + 2) (x + 2) (x - 2) (3 x + 1) = 0$

Step 2.1.15.3

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

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

Step 2.1.15.4

Add $1$ and $1$ .

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

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 $0$ .

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

$x - 2 = 0$

$3 x + 1 = 0$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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Step 2.3.2.1

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

$x + 2 = 0$

Step 2.3.2.2

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 $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 2.5.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 2.5.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 2.5.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 2.5.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 2.6

The final solution is all the values that make ${(x + 2)}^{2} (x - 2) (3 x + 1) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 2$ (Multiplicity of $2$ )

$x = 2$ (Multiplicity of $1$ )

$x = - \frac{1}{3}$ (Multiplicity of $1$ )

$x = - 2$ (Multiplicity of $2$ )

$x = 2$ (Multiplicity of $1$ )

$x = - \frac{1}{3}$ (Multiplicity of $1$ )

Step 3