Trigonometry Examples

$3$ , $125 i$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

Step 3

The root at $x = 125 i$ was found by solving for $x$ when $x - (125 i) = y$ and $y = 0$ .

The factor is $x - 125 i$

Step 4

The root at $x = - 125 i$ was found by solving for $x$ when $x - (- 125 i) = y$ and $y = 0$ .

The factor is $x + 125 i$

Step 5

Combine all the factors into a single equation.

$y = (x - 3) (x - 125 i) (x + 125 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 3) (x - 125 i) (x + 125 i)$ .

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Step 6.1

Expand $(x - 3) (x - 125 i)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x - 125 i) - 3 (x - 125 i)) (x + 125 i)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x (- 125 i) - 3 (x - 125 i)) (x + 125 i)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x (- 125 i) - 3 x - 3 (- 125 i)) (x + 125 i)$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $x$ by $x$ .

$y = (x^{2} + x (- 125 i) - 3 x - 3 (- 125 i)) (x + 125 i)$

Step 6.2.2

Multiply $- 125$ by $- 3$ .

$y = (x^{2} + x (- 125 i) - 3 x + 375 i) (x + 125 i)$

Step 6.3

Expand $(x^{2} + x (- 125 i) - 3 x + 375 i) (x + 125 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (125 i) + x (- 125 i) x + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

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

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Step 6.4.1.1.1

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

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Step 6.4.1.1.1.1

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

$y = x^{2} x + x^{2} (125 i) + x (- 125 i) x + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.1.1.2

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

$y = x^{2 + 1} + x^{2} (125 i) + x (- 125 i) x + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} (125 i) + x (- 125 i) x + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.2

Move $125$ to the left of $x^{2}$ .

$y = x^{3} + 125 \cdot (x^{2} i) + x (- 125 i) x + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.3

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

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Step 6.4.1.3.1

Move $x$ .

$y = x^{3} + 125 x^{2} i + x \cdot x (- 125 i) + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.3.2

Multiply $x$ by $x$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 125 i) (125 i) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.4

Multiply $x (- 125 i) (125 i)$ .

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Step 6.4.1.4.1

Multiply $125$ by $- 125$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 i) i - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.4.2

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 (i i)) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.4.3

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 (i i)) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.4.4

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 i^{1 + 1}) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.4.5

Add $1$ and $1$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 i^{2}) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.5

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x (- 15625 \cdot - 1) - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.6

Multiply $- 15625$ by $- 1$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + x \cdot 15625 - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.7

Move $15625$ to the left of $x$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 \cdot x - 3 x \cdot x - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.8

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

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Step 6.4.1.8.1

Move $x$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 (x \cdot x) - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.8.2

Multiply $x$ by $x$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 3 x (125 i) + 375 i x + 375 i (125 i)$

Step 6.4.1.9

Multiply $125$ by $- 3$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 375 i (125 i)$

Step 6.4.1.10

Multiply $375 i (125 i)$ .

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Step 6.4.1.10.1

Multiply $125$ by $375$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 i i$

Step 6.4.1.10.2

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 (i i)$

Step 6.4.1.10.3

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 (i i)$

Step 6.4.1.10.4

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 i^{1 + 1}$

Step 6.4.1.10.5

Add $1$ and $1$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 i^{2}$

Step 6.4.1.11

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

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x + 46875 \cdot - 1$

Step 6.4.1.12

Multiply $46875$ by $- 1$ .

$y = x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x - 46875$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Combine the opposite terms in $x^{3} + 125 x^{2} i + x^{2} (- 125 i) + 15625 x - 3 x^{2} - 375 x i + 375 i x - 46875$ .

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Step 6.4.2.1.1

Reorder the factors in the terms $125 x^{2} i$ and $x^{2} (- 125 i)$ .

$y = x^{3} + 125 i x^{2} - 125 i x^{2} + 15625 x - 3 x^{2} - 375 x i + 375 i x - 46875$

Step 6.4.2.1.2

Subtract $125 i x^{2}$ from $125 i x^{2}$ .

$y = x^{3} + 0 + 15625 x - 3 x^{2} - 375 x i + 375 i x - 46875$

Step 6.4.2.1.3

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

$y = x^{3} + 15625 x - 3 x^{2} - 375 x i + 375 i x - 46875$

Step 6.4.2.1.4

Reorder the factors in the terms $- 375 x i$ and $375 i x$ .

$y = x^{3} + 15625 x - 3 x^{2} - 375 i x + 375 i x - 46875$

Step 6.4.2.1.5

Add $- 375 i x$ and $375 i x$ .

$y = x^{3} + 15625 x - 3 x^{2} + 0 - 46875$

Step 6.4.2.1.6

Add $x^{3} + 15625 x - 3 x^{2}$ and $0$ .

$y = x^{3} + 15625 x - 3 x^{2} - 46875$

Step 6.4.2.2

Move $15625 x$ .

$y = x^{3} - 3 x^{2} + 15625 x - 46875$

Step 7