Trigonometry Examples

$\frac{3}{20} x^{4} + \frac{1}{10} - \frac{3}{10} x - \frac{1}{5} - \frac{13}{20} x + 9 x^{3}$

Step 1

Simplify each term.

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Step 1.1

Combine $\frac{3}{20}$ and $x^{4}$ .

$\frac{3 x^{4}}{20} + \frac{1}{10} - \frac{3}{10} x - \frac{1}{5} - \frac{13}{20} x + 9 x^{3}$

Step 1.2

Combine $x$ and $\frac{3}{10}$ .

$\frac{3 x^{4}}{20} + \frac{1}{10} - \frac{x \cdot 3}{10} - \frac{1}{5} - \frac{13}{20} x + 9 x^{3}$

Step 1.3

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

$\frac{3 x^{4}}{20} + \frac{1}{10} - \frac{3 \cdot x}{10} - \frac{1}{5} - \frac{13}{20} x + 9 x^{3}$

Step 1.4

Combine $x$ and $\frac{13}{20}$ .

$\frac{3 x^{4}}{20} + \frac{1}{10} - \frac{3 x}{10} - \frac{1}{5} - \frac{x \cdot 13}{20} + 9 x^{3}$

Step 1.5

Move $13$ to the left of $x$ .

$\frac{3 x^{4}}{20} + \frac{1}{10} - \frac{3 x}{10} - \frac{1}{5} - \frac{13 x}{20} + 9 x^{3}$

Step 2

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x^{4}}{20} - \frac{3 x}{10} + \frac{1}{10} - \frac{1}{5} \cdot \frac{2}{2} - \frac{13 x}{20} + 9 x^{3}$

Step 3

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

$\frac{3 x^{4}}{20} - \frac{3 x}{10} + \frac{1}{10} - \frac{2}{5 \cdot 2} - \frac{13 x}{20} + 9 x^{3}$

Step 3.2

Multiply $5$ by $2$ .

$\frac{3 x^{4}}{20} - \frac{3 x}{10} + \frac{1}{10} - \frac{2}{10} - \frac{13 x}{20} + 9 x^{3}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 x^{4}}{20} - \frac{3 x}{10} + \frac{1 - 2}{10} - \frac{13 x}{20} + 9 x^{3}$

Step 5

Subtract $2$ from $1$ .

$\frac{3 x^{4}}{20} - \frac{3 x}{10} + \frac{- 1}{10} - \frac{13 x}{20} + 9 x^{3}$

Step 6

Move the negative in front of the fraction.

$\frac{3 x^{4}}{20} - \frac{3 x}{10} - \frac{1}{10} - \frac{13 x}{20} + 9 x^{3}$

Step 7

To write $- \frac{3 x}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x^{4}}{20} - \frac{3 x}{10} \cdot \frac{2}{2} - \frac{13 x}{20} - \frac{1}{10} + 9 x^{3}$

Step 8

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{3 x}{10}$ by $\frac{2}{2}$ .

$\frac{3 x^{4}}{20} - \frac{3 x \cdot 2}{10 \cdot 2} - \frac{13 x}{20} - \frac{1}{10} + 9 x^{3}$

Step 8.2

Multiply $10$ by $2$ .

$\frac{3 x^{4}}{20} - \frac{3 x \cdot 2}{20} - \frac{13 x}{20} - \frac{1}{10} + 9 x^{3}$

Step 9

Simplify terms.

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Step 9.1

Combine the numerators over the common denominator.

$\frac{3 x^{4}}{20} + \frac{- 3 x \cdot 2 - 13 x}{20} - \frac{1}{10} + 9 x^{3}$

Step 9.2

Combine the numerators over the common denominator.

$9 x^{3} + \frac{3 x^{4} - 3 x \cdot 2 - 13 x}{20} + \frac{- 1}{10}$

Step 9.3

Multiply $2$ by $- 3$ .

$9 x^{3} + \frac{3 x^{4} - 6 x - 13 x}{20} + \frac{- 1}{10}$

Step 9.4

Subtract $13 x$ from $- 6 x$ .

$9 x^{3} + \frac{3 x^{4} - 19 x}{20} + \frac{- 1}{10}$

Step 10

Simplify each term.

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Step 10.1

Factor $x$ out of $3 x^{4} - 19 x$ .

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Step 10.1.1

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

$9 x^{3} + \frac{x (3 x^{3}) - 19 x}{20} + \frac{- 1}{10}$

Step 10.1.2

Factor $x$ out of $- 19 x$ .

$9 x^{3} + \frac{x (3 x^{3}) + x \cdot - 19}{20} + \frac{- 1}{10}$

Step 10.1.3

Factor $x$ out of $x (3 x^{3}) + x \cdot - 19$ .

$9 x^{3} + \frac{x (3 x^{3} - 19)}{20} + \frac{- 1}{10}$

Step 10.2

Move the negative in front of the fraction.

$9 x^{3} + \frac{x (3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 11

To write $9 x^{3}$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$9 x^{3} \cdot \frac{20}{20} + \frac{x (3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 12

Simplify terms.

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Step 12.1

Combine $9 x^{3}$ and $\frac{20}{20}$ .

$\frac{9 x^{3} \cdot 20}{20} + \frac{x (3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 12.2

Combine the numerators over the common denominator.

$\frac{9 x^{3} \cdot 20 + x (3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 13

Simplify the numerator.

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Step 13.1

Factor $x$ out of $9 x^{3} \cdot 20 + x (3 x^{3} - 19)$ .

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Step 13.1.1

Factor $x$ out of $9 x^{3} \cdot 20$ .

$\frac{x (9 x^{2} \cdot 20) + x (3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 13.1.2

Factor $x$ out of $x (9 x^{2} \cdot 20) + x (3 x^{3} - 19)$ .

$\frac{x (9 x^{2} \cdot 20 + 3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 13.2

Multiply $20$ by $9$ .

$\frac{x (180 x^{2} + 3 x^{3} - 19)}{20} - \frac{1}{10}$

Step 13.3

Reorder terms.

$\frac{x (3 x^{3} + 180 x^{2} - 19)}{20} - \frac{1}{10}$

Step 14

To write $- \frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x (3 x^{3} + 180 x^{2} - 19)}{20} - \frac{1}{10} \cdot \frac{2}{2}$

Step 15

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.1

Multiply $\frac{1}{10}$ by $\frac{2}{2}$ .

$\frac{x (3 x^{3} + 180 x^{2} - 19)}{20} - \frac{2}{10 \cdot 2}$

Step 15.2

Multiply $10$ by $2$ .

$\frac{x (3 x^{3} + 180 x^{2} - 19)}{20} - \frac{2}{20}$

Step 16

Combine the numerators over the common denominator.

$\frac{x (3 x^{3} + 180 x^{2} - 19) - 2}{20}$

Step 17

Simplify the numerator.

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Step 17.1

Apply the distributive property.

$\frac{x (3 x^{3}) + x (180 x^{2}) + x \cdot - 19 - 2}{20}$

Step 17.2

Simplify.

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Step 17.2.1

Rewrite using the commutative property of multiplication.

$\frac{3 x \cdot x^{3} + x (180 x^{2}) + x \cdot - 19 - 2}{20}$

Step 17.2.2

Rewrite using the commutative property of multiplication.

$\frac{3 x \cdot x^{3} + 180 x \cdot x^{2} + x \cdot - 19 - 2}{20}$

Step 17.2.3

Move $- 19$ to the left of $x$ .

$\frac{3 x \cdot x^{3} + 180 x \cdot x^{2} - 19 \cdot x - 2}{20}$

Step 17.3

Simplify each term.

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Step 17.3.1

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

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Step 17.3.1.1

Move $x^{3}$ .

$\frac{3 (x^{3} x) + 180 x \cdot x^{2} - 19 \cdot x - 2}{20}$

Step 17.3.1.2

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

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Step 17.3.1.2.1

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

$\frac{3 (x^{3} x^{1}) + 180 x \cdot x^{2} - 19 \cdot x - 2}{20}$

Step 17.3.1.2.2

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

$\frac{3 x^{3 + 1} + 180 x \cdot x^{2} - 19 \cdot x - 2}{20}$

Step 17.3.1.3

Add $3$ and $1$ .

$\frac{3 x^{4} + 180 x \cdot x^{2} - 19 \cdot x - 2}{20}$

Step 17.3.2

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

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Step 17.3.2.1

Move $x^{2}$ .

$\frac{3 x^{4} + 180 (x^{2} x) - 19 \cdot x - 2}{20}$

Step 17.3.2.2

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

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Step 17.3.2.2.1

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

$\frac{3 x^{4} + 180 (x^{2} x^{1}) - 19 \cdot x - 2}{20}$

Step 17.3.2.2.2

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

$\frac{3 x^{4} + 180 x^{2 + 1} - 19 \cdot x - 2}{20}$

Step 17.3.2.3

Add $2$ and $1$ .

$\frac{3 x^{4} + 180 x^{3} - 19 \cdot x - 2}{20}$

$\frac{3 x^{4} + 180 x^{3} - 19 x - 2}{20}$