Algebra Examples

$4$ , $2 i$

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 $x = 4$ was found by solving for $x$ when $x - (4) = y$ and $y = 0$ .

The factor is $x - 4$

Step 3

The root at $x = 2 i$ was found by solving for $x$ when $x - (2 i) = y$ and $y = 0$ .

The factor is $x - 2 i$

Step 4

The root at $x = - 2 i$ was found by solving for $x$ when $x - (- 2 i) = y$ and $y = 0$ .

The factor is $x + 2 i$

Step 5

Combine all the factors into a single equation.

$y = (x - 4) (x - 2 i) (x + 2 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x - 4) (x - 2 i) (x + 2 i)$ .

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Step 6.1

Expand $(x - 4) (x - 2 i)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x - 2 i) - 4 (x - 2 i)) (x + 2 i)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x (- 2 i) - 4 (x - 2 i)) (x + 2 i)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x (- 2 i) - 4 x - 4 (- 2 i)) (x + 2 i)$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $x$ by $x$ .

$y = (x^{2} + x (- 2 i) - 4 x - 4 (- 2 i)) (x + 2 i)$

Step 6.2.2

Multiply $- 2$ by $- 4$ .

$y = (x^{2} + x (- 2 i) - 4 x + 8 i) (x + 2 i)$

Step 6.3

Expand $(x^{2} + x (- 2 i) - 4 x + 8 i) (x + 2 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

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 $1$ .

$y = x^{2} x + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.1.1.2

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

$y = x^{2 + 1} + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} (2 i) + x (- 2 i) x + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.2

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

$y = x^{3} + 2 \cdot (x^{2} i) + x (- 2 i) x + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.3

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

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Step 6.4.1.3.1

Move $x$ .

$y = x^{3} + 2 x^{2} i + x \cdot x (- 2 i) + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.3.2

Multiply $x$ by $x$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 2 i) (2 i) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.4

Multiply $x (- 2 i) (2 i)$ .

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Step 6.4.1.4.1

Multiply $2$ by $- 2$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 i) i - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.4.2

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 (i i)) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.4.3

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 (i i)) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.4.4

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 i^{1 + 1}) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.4.5

Add $1$ and $1$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 i^{2}) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.5

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x (- 4 \cdot - 1) - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.6

Multiply $- 4$ by $- 1$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + x \cdot 4 - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.7

Move $4$ to the left of $x$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 \cdot x - 4 x \cdot x - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.8

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

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Step 6.4.1.8.1

Move $x$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 (x \cdot x) - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.8.2

Multiply $x$ by $x$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 4 x (2 i) + 8 i x + 8 i (2 i)$

Step 6.4.1.9

Multiply $2$ by $- 4$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 8 i (2 i)$

Step 6.4.1.10

Multiply $8 i (2 i)$ .

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Step 6.4.1.10.1

Multiply $2$ by $8$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 i i$

Step 6.4.1.10.2

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 (i i)$

Step 6.4.1.10.3

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 (i i)$

Step 6.4.1.10.4

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 i^{1 + 1}$

Step 6.4.1.10.5

Add $1$ and $1$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 i^{2}$

Step 6.4.1.11

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

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x + 16 \cdot - 1$

Step 6.4.1.12

Multiply $16$ by $- 1$ .

$y = x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x - 16$

Step 6.4.2

Simplify by adding terms.

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Step 6.4.2.1

Combine the opposite terms in $x^{3} + 2 x^{2} i + x^{2} (- 2 i) + 4 x - 4 x^{2} - 8 x i + 8 i x - 16$ .

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Step 6.4.2.1.1

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

$y = x^{3} + 2 i x^{2} - 2 i x^{2} + 4 x - 4 x^{2} - 8 x i + 8 i x - 16$

Step 6.4.2.1.2

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

$y = x^{3} + 0 + 4 x - 4 x^{2} - 8 x i + 8 i x - 16$

Step 6.4.2.1.3

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

$y = x^{3} + 4 x - 4 x^{2} - 8 x i + 8 i x - 16$

Step 6.4.2.1.4

Reorder the factors in the terms $- 8 x i$ and $8 i x$ .

$y = x^{3} + 4 x - 4 x^{2} - 8 i x + 8 i x - 16$

Step 6.4.2.1.5

Add $- 8 i x$ and $8 i x$ .

$y = x^{3} + 4 x - 4 x^{2} + 0 - 16$

Step 6.4.2.1.6

Add $x^{3} + 4 x - 4 x^{2}$ and $0$ .

$y = x^{3} + 4 x - 4 x^{2} - 16$

Step 6.4.2.2

Move $4 x$ .

$y = x^{3} - 4 x^{2} + 4 x - 16$

Step 7