Algebra Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine the opposite terms in $x^{m} x^{2 m} + x^{m} x^{m} + x^{m} \cdot 1 - 1 x^{2 m} - 1 x^{m} - 1 \cdot 1$ .

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Step 2.1.1

Reorder the factors in the terms $x^{m} \cdot 1$ and $- 1 x^{m}$ .

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

Step 2.1.2

Subtract $1 x^{m}$ from $1 x^{m}$ .

$x^{m} x^{2 m} + x^{m} x^{m} - 1 x^{2 m} + 0 - 1 \cdot 1$

Step 2.1.3

Add $x^{m} x^{2 m} + x^{m} x^{m} - 1 x^{2 m}$ and $0$ .

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

Step 2.2

Simplify each term.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Add $m$ and $2 m$ .

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Add $m$ and $m$ .

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

Step 2.2.3

Rewrite $- 1 x^{2 m}$ as $- x^{2 m}$ .

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

Step 2.2.4

Multiply $- 1$ by $1$ .

$x^{3 m} + x^{2 m} - x^{2 m} - 1$

Step 2.3

Combine the opposite terms in $x^{3 m} + x^{2 m} - x^{2 m} - 1$ .

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Step 2.3.1

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

$x^{3 m} + 0 - 1$

Step 2.3.2

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

$x^{3 m} - 1$