Examples

x^{3} + x^{2} + 1

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

x^{2} + 7 x + 1

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

Step 1

Multiply the expressions.

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

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

Step 2

Expand

(x^{3} + x^{2} + 1) (x^{2} + 7 x + 1)

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

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

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}

$x^{3}$ by

x^{2}

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

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Step 3.1.1.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 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

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

Step 3.1.1.2

Add

3

$3$ and

2

$2$ .

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

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

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

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

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

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

Step 3.1.3

Multiply

x^{3}

$x^{3}$ by

x

$x$ by adding the exponents.

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Step 3.1.3.1

Move

x

$x$ .

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

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

Step 3.1.3.2

Multiply

x

$x$ by

x^{3}

$x^{3}$ .

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Step 3.1.3.2.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 3.1.3.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 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

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

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

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

Step 3.1.3.3

Add

1

$1$ and

3

$3$ .

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

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

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

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

Step 3.1.4

Multiply

x^{3}

$x^{3}$ by

1

$1$ .

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

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

Step 3.1.5

Multiply

x^{2}

$x^{2}$ by

x^{2}

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

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Step 3.1.5.1

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ 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

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

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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