Examples

$x^{2} + x + 1$ , $x^{2} + 8 x + 1$

Step 1

Multiply the expressions.

$(x^{2} + x + 1) \cdot (x^{2} + 8 x + 1)$

Step 2

Expand $(x^{2} + x + 1) (x^{2} + 8 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} (8 x) + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

$x^{2 + 2} + x^{2} (8 x) + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} (8 x) + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$x^{4} + 8 x^{2} x + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.3

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

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Step 3.1.3.1

Move $x$ .

$x^{4} + 8 (x \cdot x^{2}) + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.3.2

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

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Step 3.1.3.2.1

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

$x^{4} + 8 (x^{1} x^{2}) + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.3.2.2

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

$x^{4} + 8 x^{1 + 2} + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.3.3

Add $1$ and $2$ .

$x^{4} + 8 x^{3} + x^{2} \cdot 1 + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.4

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

$x^{4} + 8 x^{3} + x^{2} + x \cdot x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.5

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

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Step 3.1.5.1

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

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Step 3.1.5.1.1

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

$x^{4} + 8 x^{3} + x^{2} + x^{1} x^{2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.5.1.2

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

$x^{4} + 8 x^{3} + x^{2} + x^{1 + 2} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.5.2

Add $1$ and $2$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + x (8 x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x \cdot x + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.7

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

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Step 3.1.7.1

Move $x$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 (x \cdot x) + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.7.2

Multiply $x$ by $x$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x^{2} + x \cdot 1 + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.8

Multiply $x$ by $1$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x^{2} + x + 1 x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.9

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

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x^{2} + x + x^{2} + 1 (8 x) + 1 \cdot 1$

Step 3.1.10

Multiply $8 x$ by $1$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x^{2} + x + x^{2} + 8 x + 1 \cdot 1$

Step 3.1.11

Multiply $1$ by $1$ .

$x^{4} + 8 x^{3} + x^{2} + x^{3} + 8 x^{2} + x + x^{2} + 8 x + 1$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $8 x^{3}$ and $x^{3}$ .

$x^{4} + 9 x^{3} + x^{2} + 8 x^{2} + x + x^{2} + 8 x + 1$

Step 3.2.2

Add $x^{2}$ and $8 x^{2}$ .

$x^{4} + 9 x^{3} + 9 x^{2} + x + x^{2} + 8 x + 1$

Step 3.2.3

Add $9 x^{2}$ and $x^{2}$ .

$x^{4} + 9 x^{3} + 10 x^{2} + x + 8 x + 1$

Step 3.2.4

Add $x$ and $8 x$ .

$x^{4} + 9 x^{3} + 10 x^{2} + 9 x + 1$

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