Trigonometry Examples

$f (x) = (x - 1) (x - \sqrt{7 i}) (x + \sqrt{7 i})$

Step 1

Expand $(x - 1) (x - \sqrt{7 i})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$f (x) = (x (x - \sqrt{7 i}) - 1 (x - \sqrt{7 i})) (x + \sqrt{7 i})$

Step 1.2

Apply the distributive property.

$f (x) = (x \cdot x + x (- \sqrt{7 i}) - 1 (x - \sqrt{7 i})) (x + \sqrt{7 i})$

Step 1.3

Apply the distributive property.

$f (x) = (x \cdot x + x (- \sqrt{7 i}) - 1 x - 1 (- \sqrt{7 i})) (x + \sqrt{7 i})$

Step 2

Simplify each term.

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Step 2.1

Multiply $x$ by $x$ .

$f (x) = (x^{2} + x (- \sqrt{7 i}) - 1 x - 1 (- \sqrt{7 i})) (x + \sqrt{7 i})$

Step 2.2

Rewrite $- 1 x$ as $- x$ .

$f (x) = (x^{2} + x (- \sqrt{7 i}) - x - 1 (- \sqrt{7 i})) (x + \sqrt{7 i})$

Step 2.3

Multiply $- 1 (- \sqrt{7 i})$ .

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Step 2.3.1

Multiply $- 1$ by $- 1$ .

$f (x) = (x^{2} + x (- \sqrt{7 i}) - x + 1 \sqrt{7 i}) (x + \sqrt{7 i})$

Step 2.3.2

Multiply $\sqrt{7 i}$ by $1$ .

$f (x) = (x^{2} + x (- \sqrt{7 i}) - x + \sqrt{7 i}) (x + \sqrt{7 i})$

Step 3

Expand $(x^{2} + x (- \sqrt{7 i}) - x + \sqrt{7 i}) (x + \sqrt{7 i})$ by multiplying each term in the first expression by each term in the second expression.

$f (x) = x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x - x \sqrt{7 i} + \sqrt{7 i} x + \sqrt{7 i} \sqrt{7 i}$

Step 4

Simplify terms.

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Step 4.1

Combine the opposite terms in $x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x - x \sqrt{7 i} + \sqrt{7 i} x + \sqrt{7 i} \sqrt{7 i}$ .

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Step 4.1.1

Reorder the factors in the terms $- x \sqrt{7 i}$ and $\sqrt{7 i} x$ .

$f (x) = x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x - x \sqrt{7 i} + x \sqrt{7 i} + \sqrt{7 i} \sqrt{7 i}$

Step 4.1.2

Add $- x \sqrt{7 i}$ and $x \sqrt{7 i}$ .

$f (x) = x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + 0 + \sqrt{7 i} \sqrt{7 i}$

Step 4.1.3

Add $x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x$ and $0$ .

$f (x) = x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

$f (x) = x^{2} x + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.1.1.2

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

$f (x) = x^{2 + 1} + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.1.2

Add $2$ and $1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x (- \sqrt{7 i}) x + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.2

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

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Step 4.2.2.1

Move $x$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x \cdot x (- \sqrt{7 i}) + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.2.2

Multiply $x$ by $x$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- \sqrt{7 i}) \sqrt{7 i} - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.3

Multiply $x (- \sqrt{7 i}) \sqrt{7 i}$ .

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Step 4.2.3.1

Raise $\sqrt{7 i}$ to the power of $1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- (\sqrt{7 i} \sqrt{7 i})) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.3.2

Raise $\sqrt{7 i}$ to the power of $1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- (\sqrt{7 i} \sqrt{7 i})) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.3.3

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

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {\sqrt{7 i}}^{1 + 1}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.3.4

Add $1$ and $1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {\sqrt{7 i}}^{2}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4

Rewrite ${\sqrt{7 i}}^{2}$ as $7 i$ .

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Step 4.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7 i}$ as ${(7 i)}^{\frac{1}{2}}$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {({(7 i)}^{\frac{1}{2}})}^{2}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {(7 i)}^{\frac{1}{2} \cdot 2}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4.3

Combine $\frac{1}{2}$ and $2$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {(7 i)}^{\frac{2}{2}}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4.4

Cancel the common factor of $2$ .

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Step 4.2.4.4.1

Cancel the common factor.

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- {(7 i)}^{\frac{2}{2}}) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4.4.2

Rewrite the expression.

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- (7 i)) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.4.5

Simplify.

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- (7 i)) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.5

Multiply $7$ by $- 1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x \cdot x + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.6

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

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Step 4.2.6.1

Move $x$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - (x \cdot x) + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.6.2

Multiply $x$ by $x$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{7 i} \sqrt{7 i}$

Step 4.2.7

Combine using the product rule for radicals.

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{7 i (7 i)}$

Step 4.2.8

Multiply $7$ by $7$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 i i}$

Step 4.2.9

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

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 (i i)}$

Step 4.2.10

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

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 (i i)}$

Step 4.2.11

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

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 i^{1 + 1}}$

Step 4.2.12

Add $1$ and $1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 i^{2}}$

Step 4.2.13

Rewrite $i^{2}$ as $- 1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{49 \cdot - 1}$

Step 4.2.14

Multiply $49$ by $- 1$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{- 49}$

Step 4.2.15

Rewrite $- 49$ as $- 1 (49)$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{- 1 \cdot 49}$

Step 4.2.16

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + \sqrt{- 1} \cdot \sqrt{49}$

Step 4.2.17

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

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + i \cdot \sqrt{49}$

Step 4.2.18

Rewrite $49$ as $7^{2}$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + i \cdot \sqrt{7^{2}}$

Step 4.2.19

Pull terms out from under the radical, assuming positive real numbers.

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + i \cdot 7$

Step 4.2.20

Move $7$ to the left of $i$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + 7 i$

Step 4.3

Simplify by adding terms.

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Step 4.3.1

Combine the opposite terms in $x^{3} + x^{2} \sqrt{7 i} + x^{2} (- \sqrt{7 i}) + x (- 7 i) - x^{2} + 7 i$ .

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Step 4.3.1.1

Reorder the factors in the terms $x^{2} \sqrt{7 i}$ and $x^{2} (- \sqrt{7 i})$ .

$f (x) = x^{3} + x^{2} \sqrt{7 i} - x^{2} \sqrt{7 i} + x (- 7 i) - x^{2} + 7 i$

Step 4.3.1.2

Subtract $x^{2} \sqrt{7 i}$ from $x^{2} \sqrt{7 i}$ .

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

Step 4.3.1.3

Add $x^{3}$ and $0$ .

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

Step 4.3.2

Reorder factors in $x^{3} + x (- 7 i) - x^{2} + 7 i$ .

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

Step 5