Algebra Examples

$(5 x - y^{2}) (25 x^{2} + 5 x y^{2} + y^{4})$

Step 1

Expand $(5 x - y^{2}) (25 x^{2} + 5 x y^{2} + y^{4})$ by multiplying each term in the first expression by each term in the second expression.

$5 x (25 x^{2}) + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite using the commutative property of multiplication.

$5 \cdot 25 x \cdot x^{2} + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.2

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

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Step 2.1.2.1

Move $x^{2}$ .

$5 \cdot 25 (x^{2} x) + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

$5 \cdot 25 (x^{2} x^{1}) + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.2.2.2

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

$5 \cdot 25 x^{2 + 1} + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.2.3

Add $2$ and $1$ .

$5 \cdot 25 x^{3} + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.3

Multiply $5$ by $25$ .

$125 x^{3} + 5 x (5 x y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.4

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

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Step 2.1.4.1

Move $x$ .

$125 x^{3} + 5 (x \cdot x) (5 y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.4.2

Multiply $x$ by $x$ .

$125 x^{3} + 5 x^{2} (5 y^{2}) + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$125 x^{3} + 5 \cdot 5 x^{2} y^{2} + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.6

Multiply $5$ by $5$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - y^{2} (25 x^{2}) - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.7

Rewrite using the commutative property of multiplication.

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 1 \cdot 25 y^{2} x^{2} - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.8

Multiply $- 1$ by $25$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - y^{2} (5 x y^{2}) - y^{2} y^{4}$

Step 2.1.9

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

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Step 2.1.9.1

Move $y^{2}$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - (y^{2} y^{2}) (5 x) - y^{2} y^{4}$

Step 2.1.9.2

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

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - y^{2 + 2} (5 x) - y^{2} y^{4}$

Step 2.1.9.3

Add $2$ and $2$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - y^{4} (5 x) - y^{2} y^{4}$

Step 2.1.10

Rewrite using the commutative property of multiplication.

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 1 \cdot 5 y^{4} x - y^{2} y^{4}$

Step 2.1.11

Multiply $- 1$ by $5$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 5 y^{4} x - y^{2} y^{4}$

Step 2.1.12

Multiply $y^{2}$ by $y^{4}$ by adding the exponents.

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Step 2.1.12.1

Move $y^{4}$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 5 y^{4} x - (y^{4} y^{2})$

Step 2.1.12.2

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

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 5 y^{4} x - y^{4 + 2}$

Step 2.1.12.3

Add $4$ and $2$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 5 y^{4} x - y^{6}$

Step 2.2

Combine the opposite terms in $125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 y^{2} x^{2} - 5 y^{4} x - y^{6}$ .

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Step 2.2.1

Reorder the factors in the terms $25 x^{2} y^{2}$ and $- 25 y^{2} x^{2}$ .

$125 x^{3} + 25 x^{2} y^{2} + 5 x y^{4} - 25 x^{2} y^{2} - 5 y^{4} x - y^{6}$

Step 2.2.2

Subtract $25 x^{2} y^{2}$ from $25 x^{2} y^{2}$ .

$125 x^{3} + 5 x y^{4} + 0 - 5 y^{4} x - y^{6}$

Step 2.2.3

Add $125 x^{3} + 5 x y^{4}$ and $0$ .

$125 x^{3} + 5 x y^{4} - 5 y^{4} x - y^{6}$

Step 2.2.4

Reorder the factors in the terms $5 x y^{4}$ and $- 5 y^{4} x$ .

$125 x^{3} + 5 y^{4} x - 5 y^{4} x - y^{6}$

Step 2.2.5

Subtract $5 y^{4} x$ from $5 y^{4} x$ .

$125 x^{3} + 0 - y^{6}$

Step 2.2.6

Add $125 x^{3}$ and $0$ .

$125 x^{3} - y^{6}$