Finite Math Examples

$(x + 1) (x - 2) (x - i) (x + i)$

Step 1

Expand $(x + 1) (x - 2)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$(x (x - 2) + 1 (x - 2)) (x - i) (x + i)$

Step 1.2

Apply the distributive property.

$(x \cdot x + x \cdot - 2 + 1 (x - 2)) (x - i) (x + i)$

Step 1.3

Apply the distributive property.

$(x \cdot x + x \cdot - 2 + 1 x + 1 \cdot - 2) (x - i) (x + i)$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by $x$ .

$(x^{2} + x \cdot - 2 + 1 x + 1 \cdot - 2) (x - i) (x + i)$

Step 2.1.2

Move $- 2$ to the left of $x$ .

$(x^{2} - 2 \cdot x + 1 x + 1 \cdot - 2) (x - i) (x + i)$

Step 2.1.3

Multiply $x$ by $1$ .

$(x^{2} - 2 x + x + 1 \cdot - 2) (x - i) (x + i)$

Step 2.1.4

Multiply $- 2$ by $1$ .

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

Step 2.2

Add $- 2 x$ and $x$ .

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

Step 3

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

$(x^{2} x + x^{2} (- i) - x \cdot x - x (- i) - 2 x - 2 (- i)) (x + i)$

Step 4

Simplify each term.

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Step 4.1

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

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Step 4.1.1

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

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Step 4.1.1.1

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

$(x^{2} x^{1} + x^{2} (- i) - x \cdot x - x (- i) - 2 x - 2 (- i)) (x + i)$

Step 4.1.1.2

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

$(x^{2 + 1} + x^{2} (- i) - x \cdot x - x (- i) - 2 x - 2 (- i)) (x + i)$

Step 4.1.2

Add $2$ and $1$ .

$(x^{3} + x^{2} (- i) - x \cdot x - x (- i) - 2 x - 2 (- i)) (x + i)$

Step 4.2

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

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Step 4.2.1

Move $x$ .

$(x^{3} + x^{2} (- i) - (x \cdot x) - x (- i) - 2 x - 2 (- i)) (x + i)$

Step 4.2.2

Multiply $x$ by $x$ .

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

Step 4.3

Multiply $- x (- i)$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$(x^{3} + x^{2} (- i) - x^{2} + 1 x i - 2 x - 2 (- i)) (x + i)$

Step 4.3.2

Multiply $x$ by $1$ .

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

Step 4.4

Multiply $- 1$ by $- 2$ .

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

Step 5

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

$x^{3} x + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x - 2 x i + 2 i x + 2 i i$

Step 6

Simplify terms.

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Step 6.1

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

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Step 6.1.1

Reorder the factors in the terms $- 2 x i$ and $2 i x$ .

$x^{3} x + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x - 2 i x + 2 i x + 2 i i$

Step 6.1.2

Add $- 2 i x$ and $2 i x$ .

$x^{3} x + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x + 0 + 2 i i$

Step 6.1.3

Add $x^{3} x + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x$ and $0$ .

$x^{3} x + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x + 2 i i$

Step 6.2

Simplify each term.

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Step 6.2.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 6.2.1.1

Multiply $x^{3}$ by $x$ .

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Step 6.2.1.1.1

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

$x^{3} x^{1} + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x + 2 i i$

Step 6.2.1.1.2

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

$x^{3 + 1} + x^{3} i + x^{2} (- i) x + x^{2} (- i) i - x^{2} x - x^{2} i + x i x + x i i - 2 x \cdot x + 2 i i$

Step 6.2.1.2

Add $3$ and $1$ .

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

Step 6.2.2

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

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Step 6.2.2.1

Move $x$ .

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

Step 6.2.2.2

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

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Step 6.2.2.2.1

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

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

Step 6.2.2.2.2

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

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

Step 6.2.2.3

Add $1$ and $2$ .

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

Step 6.2.3

Multiply $x^{2} (- i) i$ .

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Step 6.2.3.1

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

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

Step 6.2.3.2

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

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

Step 6.2.3.3

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

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

Step 6.2.3.4

Add $1$ and $1$ .

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

Step 6.2.4

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

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

Step 6.2.5

Multiply $- 1$ by $- 1$ .

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

Step 6.2.6

Multiply $x^{2}$ by $1$ .

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

Step 6.2.7

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

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Step 6.2.7.1

Move $x$ .

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

Step 6.2.7.2

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

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Step 6.2.7.2.1

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

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

Step 6.2.7.2.2

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

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

Step 6.2.7.3

Add $1$ and $2$ .

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

Step 6.2.8

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

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Step 6.2.8.1

Move $x$ .

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

Step 6.2.8.2

Multiply $x$ by $x$ .

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

Step 6.2.9

Multiply $x i i$ .

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Step 6.2.9.1

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

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

Step 6.2.9.2

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

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

Step 6.2.9.3

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

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

Step 6.2.9.4

Add $1$ and $1$ .

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

Step 6.2.10

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

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

Step 6.2.11

Move $- 1$ to the left of $x$ .

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

Step 6.2.12

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

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

Step 6.2.13

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

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Step 6.2.13.1

Move $x$ .

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

Step 6.2.13.2

Multiply $x$ by $x$ .

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

Step 6.2.14

Multiply $2 i i$ .

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Step 6.2.14.1

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

$x^{4} + x^{3} i + x^{3} (- i) + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} + 2 (i^{1} i)$

Step 6.2.14.2

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

$x^{4} + x^{3} i + x^{3} (- i) + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} + 2 (i^{1} i^{1})$

Step 6.2.14.3

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

$x^{4} + x^{3} i + x^{3} (- i) + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} + 2 i^{1 + 1}$

Step 6.2.14.4

Add $1$ and $1$ .

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

Step 6.2.15

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

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

Step 6.2.16

Multiply $2$ by $- 1$ .

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

Step 6.3

Simplify by adding terms.

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Step 6.3.1

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

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Step 6.3.1.1

Reorder the factors in the terms $x^{3} i$ and $x^{3} (- i)$ .

$x^{4} + i x^{3} - i x^{3} + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} - 2$

Step 6.3.1.2

Subtract $i x^{3}$ from $i x^{3}$ .

$x^{4} + 0 + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} - 2$

Step 6.3.1.3

Add $x^{4}$ and $0$ .

$x^{4} + x^{2} - x^{3} - x^{2} i + x^{2} i - x - 2 x^{2} - 2$

Step 6.3.1.4

Add $- x^{2} i$ and $x^{2} i$ .

$x^{4} + x^{2} - x^{3} + 0 - x - 2 x^{2} - 2$

Step 6.3.1.5

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

$x^{4} + x^{2} - x^{3} - x - 2 x^{2} - 2$

Step 6.3.2

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

$x^{4} - x^{2} - x^{3} - x - 2$

Step 6.3.3

Move $- x^{2}$ .

$x^{4} - x^{3} - x^{2} - x - 2$