Trigonometry Examples

$\frac{x}{x^{3} + 15 x^{2} + 56 x} = \frac{0}{x (x + 4) (x + 7) x + (x + 8)}$

Step 1

Factor each term.

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Step 1.1

Factor $x$ out of $x^{3} + 15 x^{2} + 56 x$ .

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Step 1.1.1

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

$\frac{x}{x \cdot x^{2} + 15 x^{2} + 56 x} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.1.2

Factor $x$ out of $15 x^{2}$ .

$\frac{x}{x \cdot x^{2} + x (15 x) + 56 x} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.1.3

Factor $x$ out of $56 x$ .

$\frac{x}{x \cdot x^{2} + x (15 x) + x \cdot 56} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.1.4

Factor $x$ out of $x \cdot x^{2} + x (15 x)$ .

$\frac{x}{x (x^{2} + 15 x) + x \cdot 56} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.1.5

Factor $x$ out of $x (x^{2} + 15 x) + x \cdot 56$ .

$\frac{x}{x (x^{2} + 15 x + 56)} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.2

Factor.

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Step 1.2.1

Factor $x^{2} + 15 x + 56$ using the AC method.

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Step 1.2.1.1

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 $56$ and whose sum is $15$ .

$7, 8$

Step 1.2.1.2

Write the factored form using these integers.

$\frac{x}{x ((x + 7) (x + 8))} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.2.2

Remove unnecessary parentheses.

$\frac{x}{x (x + 7) (x + 8)} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.3

Reduce the expression $\frac{x}{x (x + 7) (x + 8)}$ by cancelling the common factors.

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Step 1.3.1

Cancel the common factor.

$\frac{x}{x (x + 7) (x + 8)} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.3.2

Rewrite the expression.

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x (x + 4) (x + 7) x + x + 8}$

Step 1.4

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

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Step 1.4.1

Move $x$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x \cdot x (x + 4) (x + 7) + x + 8}$

Step 1.4.2

Multiply $x$ by $x$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{2} (x + 4) (x + 7) + x + 8}$

Step 1.5

Apply the distributive property.

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{(x^{2} x + x^{2} \cdot 4) (x + 7) + x + 8}$

Step 1.6

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

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Step 1.6.1

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

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Step 1.6.1.1

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{(x^{2} x^{1} + x^{2} \cdot 4) (x + 7) + x + 8}$

Step 1.6.1.2

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{(x^{2 + 1} + x^{2} \cdot 4) (x + 7) + x + 8}$

Step 1.6.2

Add $2$ and $1$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{(x^{3} + x^{2} \cdot 4) (x + 7) + x + 8}$

Step 1.7

Move $4$ to the left of $x^{2}$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{(x^{3} + 4 x^{2}) (x + 7) + x + 8}$

Step 1.8

Expand $(x^{3} + 4 x^{2}) (x + 7)$ using the FOIL Method.

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Step 1.8.1

Apply the distributive property.

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{3} (x + 7) + 4 x^{2} (x + 7) + x + 8}$

Step 1.8.2

Apply the distributive property.

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{3} x + x^{3} \cdot 7 + 4 x^{2} (x + 7) + x + 8}$

Step 1.8.3

Apply the distributive property.

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{3} x + x^{3} \cdot 7 + 4 x^{2} x + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9

Simplify and combine like terms.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 1.9.1.1.1

Multiply $x^{3}$ by $x$ .

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Step 1.9.1.1.1.1

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{3} x^{1} + x^{3} \cdot 7 + 4 x^{2} x + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.1.1.2

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{3 + 1} + x^{3} \cdot 7 + 4 x^{2} x + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.1.2

Add $3$ and $1$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + x^{3} \cdot 7 + 4 x^{2} x + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.2

Move $7$ to the left of $x^{3}$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 x^{2} x + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.3

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

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Step 1.9.1.3.1

Move $x$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 (x \cdot x^{2}) + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.3.2

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

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Step 1.9.1.3.2.1

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 (x^{1} x^{2}) + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.3.2.2

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

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 x^{1 + 2} + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.3.3

Add $1$ and $2$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 x^{3} + 4 x^{2} \cdot 7 + x + 8}$

Step 1.9.1.4

Multiply $7$ by $4$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 7 x^{3} + 4 x^{3} + 28 x^{2} + x + 8}$

Step 1.9.2

Add $7 x^{3}$ and $4 x^{3}$ .

$\frac{1}{(x + 7) (x + 8)} = \frac{0}{x^{4} + 11 x^{3} + 28 x^{2} + x + 8}$

Step 1.10

Divide $0$ by $x^{4} + 11 x^{3} + 28 x^{2} + x + 8$ .

$\frac{1}{(x + 7) (x + 8)} = 0$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(x + 7) (x + 8), 1$

Step 2.2

The LCM of one and any expression is the expression.

$(x + 7) (x + 8)$

Step 3

Multiply each term in $\frac{1}{(x + 7) (x + 8)} = 0$ by $(x + 7) (x + 8)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{(x + 7) (x + 8)} = 0$ by $(x + 7) (x + 8)$ .

$\frac{1}{(x + 7) (x + 8)} ((x + 7) (x + 8)) = 0 ((x + 7) (x + 8))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(x + 7) (x + 8)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{1}{(x + 7) (x + 8)} ((x + 7) (x + 8)) = 0 ((x + 7) (x + 8))$

Step 3.2.1.2

Rewrite the expression.

$1 = 0 ((x + 7) (x + 8))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Expand $(x + 7) (x + 8)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$1 = 0 (x (x + 8) + 7 (x + 8))$

Step 3.3.1.2

Apply the distributive property.

$1 = 0 (x \cdot x + x \cdot 8 + 7 (x + 8))$

Step 3.3.1.3

Apply the distributive property.

$1 = 0 (x \cdot x + x \cdot 8 + 7 x + 7 \cdot 8)$

Step 3.3.2

Simplify and combine like terms.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $x$ by $x$ .

$1 = 0 (x^{2} + x \cdot 8 + 7 x + 7 \cdot 8)$

Step 3.3.2.1.2

Move $8$ to the left of $x$ .

$1 = 0 (x^{2} + 8 \cdot x + 7 x + 7 \cdot 8)$

Step 3.3.2.1.3

Multiply $7$ by $8$ .

$1 = 0 (x^{2} + 8 x + 7 x + 56)$

Step 3.3.2.2

Add $8 x$ and $7 x$ .

$1 = 0 (x^{2} + 15 x + 56)$

Step 3.3.3

Multiply $0$ by $x^{2} + 15 x + 56$ .

$1 = 0$

Step 4

Since $1 \neq 0$ , there are no solutions.

No solution