Algebra Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $1$ by $1$ .

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

Step 2.1.2

Multiply $- x$ by $1$ .

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

Step 2.1.3

Multiply $x$ by $1$ .

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

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

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

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Step 2.1.5.1

Move $x$ .

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

Step 2.1.5.2

Multiply $x$ by $x$ .

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

Step 2.2

Add $- x$ and $x$ .

$(1 + 0 - x^{2}) (1 + x^{2})$

Step 2.3

Add $1$ and $0$ .

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

Step 3

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

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

$1 \cdot 1 + 1 x^{2} - x^{2} \cdot 1 - x^{2} x^{2}$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $1$ by $1$ .

$1 + 1 x^{2} - x^{2} \cdot 1 - x^{2} x^{2}$

Step 4.1.2

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

$1 + x^{2} - x^{2} \cdot 1 - x^{2} x^{2}$

Step 4.1.3

Multiply $- 1$ by $1$ .

$1 + x^{2} - x^{2} - x^{2} x^{2}$

Step 4.1.4

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

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Step 4.1.4.1

Move $x^{2}$ .

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

Step 4.1.4.2

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

$1 + x^{2} - x^{2} - x^{2 + 2}$

Step 4.1.4.3

Add $2$ and $2$ .

$1 + x^{2} - x^{2} - x^{4}$

Step 4.2

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

$1 + 0 - x^{4}$

Step 4.3

Add $1$ and $0$ .

$1 - x^{4}$