Examples

$x^{3} + x^{2} + 1$ , $x^{2} + 7 x + 1$

Step 1

Multiply the expressions.

$(x^{3} + x^{2} + 1) \cdot (x^{2} + 7 x + 1)$

Step 2

Expand $(x^{3} + x^{2} + 1) (x^{2} + 7 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} x^{2} + x^{3} (7 x) + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x^{3}$ 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.

$x^{3 + 2} + x^{3} (7 x) + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.1.2

Add $3$ and $2$ .

$x^{5} + x^{3} (7 x) + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.2

Rewrite using the commutative property of multiplication.

$x^{5} + 7 x^{3} x + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.3

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

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Step 3.1.3.1

Move $x$ .

$x^{5} + 7 (x \cdot x^{3}) + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.3.2

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

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Step 3.1.3.2.1

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

$x^{5} + 7 (x^{1} x^{3}) + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.3.2.2

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

$x^{5} + 7 x^{1 + 3} + x^{3} \cdot 1 + x^{2} x^{2} + x^{2} (7 x) + x^{2} \cdot 1 + 1 x^{2} + 1 (7 x) + 1 \cdot 1$

Step 3.1.3.3

Add $1$ and $3$ .

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

Step 3.1.4

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

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

Add $2$ and $2$ .

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

Step 3.1.6

Rewrite using the commutative property of multiplication.

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

Step 3.1.7

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

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Step 3.1.7.1

Move $x$ .

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

Step 3.1.7.2

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

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Step 3.1.7.2.1

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

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

Step 3.1.7.2.2

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

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

Step 3.1.7.3

Add $1$ and $2$ .

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

Step 3.1.8

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

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

Step 3.1.9

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

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

Step 3.1.10

Multiply $7 x$ by $1$ .

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

Step 3.1.11

Multiply $1$ by $1$ .

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

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $7 x^{4}$ and $x^{4}$ .

$x^{5} + 8 x^{4} + x^{3} + 7 x^{3} + x^{2} + x^{2} + 7 x + 1$

Step 3.2.2

Add $x^{3}$ and $7 x^{3}$ .

$x^{5} + 8 x^{4} + 8 x^{3} + x^{2} + x^{2} + 7 x + 1$

Step 3.2.3

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

$x^{5} + 8 x^{4} + 8 x^{3} + 2 x^{2} + 7 x + 1$

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