Simplify (2x-y)(4x^2+2xy+y^2)

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

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $2$ by $4$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $2$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 1$ by $4$ .

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

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $- 1$ by $2$ .

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

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

Move $y^{2}$ .

$8 x^{3} + 4 x^{2} y + 2 x y^{2} - 4 y x^{2} - 2 y^{2} x - (y^{2} y)$

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

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

$8 x^{3} + 4 x^{2} y + 2 x y^{2} - 4 y x^{2} - 2 y^{2} x - (y^{2} y^{1})$

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

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

Add $2$ and $1$ .

$8 x^{3} + 4 x^{2} y + 2 x y^{2} - 4 y x^{2} - 2 y^{2} x - y^{3}$

Combine the opposite terms in $8 x^{3} + 4 x^{2} y + 2 x y^{2} - 4 y x^{2} - 2 y^{2} x - y^{3}$ .

Reorder the factors in the terms $4 x^{2} y$ and $- 4 y x^{2}$ .

$8 x^{3} + 4 x^{2} y + 2 x y^{2} - 4 x^{2} y - 2 y^{2} x - y^{3}$

Subtract $4 x^{2} y$ from $4 x^{2} y$ .

$8 x^{3} + 2 x y^{2} + 0 - 2 y^{2} x - y^{3}$

Add $8 x^{3} + 2 x y^{2}$ and $0$ .

$8 x^{3} + 2 x y^{2} - 2 y^{2} x - y^{3}$

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

$8 x^{3} + 2 y^{2} x - 2 y^{2} x - y^{3}$

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

$8 x^{3} + 0 - y^{3}$

Add $8 x^{3}$ and $0$ .

$8 x^{3} - y^{3}$

Simplify (2x-y)(4x^2+2xy+y^2)

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