Algebra Examples

$(x - x^{3} + x^{5}) (x^{2} - 1 + x^{4})$

Step 1

Expand $(x - x^{3} + x^{5}) (x^{2} - 1 + x^{4})$ by multiplying each term in the first expression by each term in the second expression.

$x \cdot x^{2} + x \cdot - 1 + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

$x^{1} x^{2} + x \cdot - 1 + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.1.1.2

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

$x^{1 + 2} + x \cdot - 1 + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.1.2

Add $1$ and $2$ .

$x^{3} + x \cdot - 1 + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.2

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

$x^{3} - 1 \cdot x + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.3

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

$x^{3} - x + x \cdot x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.4

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 2.1.4.1

Multiply $x$ by $x^{4}$ .

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Step 2.1.4.1.1

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

$x^{3} - x + x^{1} x^{4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.4.1.2

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

$x^{3} - x + x^{1 + 4} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.4.2

Add $1$ and $4$ .

$x^{3} - x + x^{5} - x^{3} x^{2} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.5

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.1.5.1

Move $x^{2}$ .

$x^{3} - x + x^{5} - (x^{2} x^{3}) - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.5.2

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

$x^{3} - x + x^{5} - x^{2 + 3} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.5.3

Add $2$ and $3$ .

$x^{3} - x + x^{5} - x^{5} - x^{3} \cdot - 1 - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.6

Multiply $- x^{3} \cdot - 1$ .

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Step 2.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{3} - x + x^{5} - x^{5} + 1 x^{3} - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.6.2

Multiply $x^{3}$ by $1$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{3} x^{4} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.7

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 2.1.7.1

Move $x^{4}$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - (x^{4} x^{3}) + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.7.2

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

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{4 + 3} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.7.3

Add $4$ and $3$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{5} x^{2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.8

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 2.1.8.1

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

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{5 + 2} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.8.2

Add $5$ and $2$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} + x^{5} \cdot - 1 + x^{5} x^{4}$

Step 2.1.9

Move $- 1$ to the left of $x^{5}$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} - 1 \cdot x^{5} + x^{5} x^{4}$

Step 2.1.10

Rewrite $- 1 x^{5}$ as $- x^{5}$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} - x^{5} + x^{5} x^{4}$

Step 2.1.11

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 2.1.11.1

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

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} - x^{5} + x^{5 + 4}$

Step 2.1.11.2

Add $5$ and $4$ .

$x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} - x^{5} + x^{9}$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Combine the opposite terms in $x^{3} - x + x^{5} - x^{5} + x^{3} - x^{7} + x^{7} - x^{5} + x^{9}$ .

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Step 2.2.1.1

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

$x^{3} - x + 0 + x^{3} - x^{7} + x^{7} - x^{5} + x^{9}$

Step 2.2.1.2

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

$x^{3} - x + x^{3} - x^{7} + x^{7} - x^{5} + x^{9}$

Step 2.2.1.3

Add $- x^{7}$ and $x^{7}$ .

$x^{3} - x + x^{3} + 0 - x^{5} + x^{9}$

Step 2.2.1.4

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

$x^{3} - x + x^{3} - x^{5} + x^{9}$

Step 2.2.2

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

$2 x^{3} - x - x^{5} + x^{9}$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Move $- x$ .

$2 x^{3} - x^{5} + x^{9} - x$

Step 2.2.3.2

Move $2 x^{3}$ .

$- x^{5} + x^{9} + 2 x^{3} - x$

Step 2.2.3.3

Reorder $- x^{5}$ and $x^{9}$ .

$x^{9} - x^{5} + 2 x^{3} - x$