Algebra Examples

$5 x^{2} + x$ and $4 x^{3} + 2 x^{2} - 4 x$

Step 1

Multiply the expressions.

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.2

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

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Step 3.1.2.1

Move $x^{3}$ .

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Add $3$ and $2$ .

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

Step 3.1.3

Multiply $5$ by $4$ .

$20 x^{5} + 5 x^{2} (2 x^{2}) + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$20 x^{5} + 5 \cdot 2 x^{2} x^{2} + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.5

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

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Step 3.1.5.1

Move $x^{2}$ .

$20 x^{5} + 5 \cdot 2 (x^{2} x^{2}) + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.5.2

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

$20 x^{5} + 5 \cdot 2 x^{2 + 2} + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.5.3

Add $2$ and $2$ .

$20 x^{5} + 5 \cdot 2 x^{4} + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.6

Multiply $5$ by $2$ .

$20 x^{5} + 10 x^{4} + 5 x^{2} (- 4 x) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.7

Rewrite using the commutative property of multiplication.

$20 x^{5} + 10 x^{4} + 5 \cdot - 4 x^{2} x + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.8

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

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Step 3.1.8.1

Move $x$ .

$20 x^{5} + 10 x^{4} + 5 \cdot - 4 (x \cdot x^{2}) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.8.2

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

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Step 3.1.8.2.1

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

$20 x^{5} + 10 x^{4} + 5 \cdot - 4 (x^{1} x^{2}) + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.8.2.2

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

$20 x^{5} + 10 x^{4} + 5 \cdot - 4 x^{1 + 2} + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.8.3

Add $1$ and $2$ .

$20 x^{5} + 10 x^{4} + 5 \cdot - 4 x^{3} + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.9

Multiply $5$ by $- 4$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + x (4 x^{3}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.10

Rewrite using the commutative property of multiplication.

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x \cdot x^{3} + x (2 x^{2}) + x (- 4 x)$

Step 3.1.11

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

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Step 3.1.11.1

Move $x^{3}$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 (x^{3} x) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.11.2

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

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Step 3.1.11.2.1

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

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 (x^{3} x^{1}) + x (2 x^{2}) + x (- 4 x)$

Step 3.1.11.2.2

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

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{3 + 1} + x (2 x^{2}) + x (- 4 x)$

Step 3.1.11.3

Add $3$ and $1$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + x (2 x^{2}) + x (- 4 x)$

Step 3.1.12

Rewrite using the commutative property of multiplication.

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x \cdot x^{2} + x (- 4 x)$

Step 3.1.13

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

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Step 3.1.13.1

Move $x^{2}$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 (x^{2} x) + x (- 4 x)$

Step 3.1.13.2

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

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Step 3.1.13.2.1

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

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 (x^{2} x^{1}) + x (- 4 x)$

Step 3.1.13.2.2

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

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x^{2 + 1} + x (- 4 x)$

Step 3.1.13.3

Add $2$ and $1$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x^{3} + x (- 4 x)$

Step 3.1.14

Rewrite using the commutative property of multiplication.

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x^{3} - 4 x \cdot x$

Step 3.1.15

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

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Step 3.1.15.1

Move $x$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x^{3} - 4 (x \cdot x)$

Step 3.1.15.2

Multiply $x$ by $x$ .

$20 x^{5} + 10 x^{4} - 20 x^{3} + 4 x^{4} + 2 x^{3} - 4 x^{2}$

Step 3.2

Simplify by adding terms.

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Step 3.2.1

Add $10 x^{4}$ and $4 x^{4}$ .

$20 x^{5} + 14 x^{4} - 20 x^{3} + 2 x^{3} - 4 x^{2}$

Step 3.2.2

Add $- 20 x^{3}$ and $2 x^{3}$ .

$20 x^{5} + 14 x^{4} - 18 x^{3} - 4 x^{2}$