Trigonometry Examples

$- i$

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

The factor is $x + i$

Step 3

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

The factor is $x - i$

Step 4

Combine all the factors into a single equation.

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

Step 5

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

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify terms.

Tap for more steps...

Step 5.2.1

Combine the opposite terms in $x \cdot x + x (- i) + i x + i (- i)$ .

Tap for more steps...

Step 5.2.1.1

Reorder the factors in the terms $x (- i)$ and $i x$ .

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

Step 5.2.1.2

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

$y = x \cdot x + 0 + i (- i)$

Step 5.2.1.3

Add $x \cdot x$ and $0$ .

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

Step 5.2.2

Simplify each term.

Tap for more steps...

Step 5.2.2.1

Multiply $x$ by $x$ .

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

Step 5.2.2.2

Multiply $i (- i)$ .

Tap for more steps...

Step 5.2.2.2.1

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

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

Step 5.2.2.2.2

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

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

Step 5.2.2.2.3

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

$y = x^{2} - i^{1 + 1}$

Step 5.2.2.2.4

Add $1$ and $1$ .

$y = x^{2} - i^{2}$

Step 5.2.2.3

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

$y = x^{2} + 1$

Step 5.2.2.4

Multiply $- 1$ by $- 1$ .

$y = x^{2} + 1$

Step 6