Algebra Examples

$- \sqrt{2}$ , $5 i$

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 $x = - \sqrt{2}$ was found by solving for $x$ when $x - (- \sqrt{2}) = y$ and $y = 0$ .

The factor is $x + \sqrt{2}$

Step 3

The root at $x = 5 i$ was found by solving for $x$ when $x - (5 i) = y$ and $y = 0$ .

The factor is $x - 5 i$

Step 4

The root at $x = - 5 i$ was found by solving for $x$ when $x - (- 5 i) = y$ and $y = 0$ .

The factor is $x + 5 i$

Step 5

Combine all the factors into a single equation.

$y = (x + \sqrt{2}) (x - 5 i) (x + 5 i)$

Step 6

Multiply all the factors to simplify the equation $y = (x + \sqrt{2}) (x - 5 i) (x + 5 i)$ .

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Step 6.1

Expand $(x + \sqrt{2}) (x - 5 i)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$y = (x (x - 5 i) + \sqrt{2} (x - 5 i)) (x + 5 i)$

Step 6.1.2

Apply the distributive property.

$y = (x \cdot x + x (- 5 i) + \sqrt{2} (x - 5 i)) (x + 5 i)$

Step 6.1.3

Apply the distributive property.

$y = (x \cdot x + x (- 5 i) + \sqrt{2} x + \sqrt{2} (- 5 i)) (x + 5 i)$

Step 6.2

Multiply $x$ by $x$ .

$y = (x^{2} + x (- 5 i) + \sqrt{2} x + \sqrt{2} (- 5 i)) (x + 5 i)$

Step 6.3

Expand $(x^{2} + x (- 5 i) + \sqrt{2} x + \sqrt{2} (- 5 i)) (x + 5 i)$ by multiplying each term in the first expression by each term in the second expression.

$y = x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} x (5 i) + \sqrt{2} (- 5 i) x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4

Simplify terms.

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Step 6.4.1

Combine the opposite terms in $x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} x (5 i) + \sqrt{2} (- 5 i) x + \sqrt{2} (- 5 i) (5 i)$ .

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Step 6.4.1.1

Reorder the factors in the terms $\sqrt{2} x (5 i)$ and $\sqrt{2} (- 5 i) x$ .

$y = x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + 5 i x \sqrt{2} - 5 i x \sqrt{2} + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.1.2

Subtract $5 i x \sqrt{2}$ from $5 i x \sqrt{2}$ .

$y = x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + 0 + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.1.3

Add $x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x$ and $0$ .

$y = x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2

Simplify each term.

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Step 6.4.2.1

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

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Step 6.4.2.1.1

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

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Step 6.4.2.1.1.1

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

$y = x^{2} x + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.1.1.2

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

$y = x^{2 + 1} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.1.2

Add $2$ and $1$ .

$y = x^{3} + x^{2} (5 i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.2

Move $5$ to the left of $x^{2}$ .

$y = x^{3} + 5 \cdot (x^{2} i) + x (- 5 i) x + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.3

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

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Step 6.4.2.3.1

Move $x$ .

$y = x^{3} + 5 x^{2} i + x \cdot x (- 5 i) + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.3.2

Multiply $x$ by $x$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 5 i) (5 i) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.4

Multiply $x (- 5 i) (5 i)$ .

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Step 6.4.2.4.1

Multiply $5$ by $- 5$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i) i + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.4.2

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 (i i)) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.4.3

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 (i i)) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.4.4

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i^{1 + 1}) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.4.5

Add $1$ and $1$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 i^{2}) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.5

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x (- 25 \cdot - 1) + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.6

Multiply $- 25$ by $- 1$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + x \cdot 25 + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.7

Move $25$ to the left of $x$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 \cdot x + \sqrt{2} x \cdot x + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.8

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

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Step 6.4.2.8.1

Move $x$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} (x \cdot x) + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.8.2

Multiply $x$ by $x$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 5 i) (5 i)$

Step 6.4.2.9

Multiply $\sqrt{2} (- 5 i) (5 i)$ .

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Step 6.4.2.9.1

Multiply $5$ by $- 5$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 i) i$

Step 6.4.2.9.2

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 (i i))$

Step 6.4.2.9.3

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 (i i))$

Step 6.4.2.9.4

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 i^{1 + 1})$

Step 6.4.2.9.5

Add $1$ and $1$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 i^{2})$

Step 6.4.2.10

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

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} (- 25 \cdot - 1)$

Step 6.4.2.11

Multiply $- 25$ by $- 1$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + \sqrt{2} \cdot 25$

Step 6.4.2.12

Move $25$ to the left of $\sqrt{2}$ .

$y = x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + 25 \sqrt{2}$

Step 6.4.3

Simplify by adding terms.

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Step 6.4.3.1

Combine the opposite terms in $x^{3} + 5 x^{2} i + x^{2} (- 5 i) + 25 x + \sqrt{2} x^{2} + 25 \sqrt{2}$ .

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Step 6.4.3.1.1

Reorder the factors in the terms $5 x^{2} i$ and $x^{2} (- 5 i)$ .

$y = x^{3} + 5 i x^{2} - 5 i x^{2} + 25 x + \sqrt{2} x^{2} + 25 \sqrt{2}$

Step 6.4.3.1.2

Subtract $5 i x^{2}$ from $5 i x^{2}$ .

$y = x^{3} + 0 + 25 x + \sqrt{2} x^{2} + 25 \sqrt{2}$

Step 6.4.3.1.3

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

$y = x^{3} + 25 x + \sqrt{2} x^{2} + 25 \sqrt{2}$

Step 6.4.3.2

Move $25 x$ .

$y = x^{3} + \sqrt{2} x^{2} + 25 x + 25 \sqrt{2}$

Step 7