Simplify (a-b)(a^2+b^2+ab)(a^2+b^2)

$(a - b) (a^{2} + b^{2} + a b) (a^{2} + b^{2})$

Step 1

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

$(a \cdot a^{2} + a b^{2} + a (a b) - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

Step 2

Simplify terms.

Simplify each term.

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

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

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

$(a^{1} a^{2} + a b^{2} + a (a b) - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

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

$(a^{1 + 2} + a b^{2} + a (a b) - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

Add $1$ and $2$ .

$(a^{3} + a b^{2} + a (a b) - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

$(a^{3} + a b^{2} + a \cdot a b - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

Multiply $a$ by $a$ .

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - b \cdot b^{2} - b (a b)) (a^{2} + b^{2})$

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

Move $b^{2}$ .

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - (b^{2} b) - b (a b)) (a^{2} + b^{2})$

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

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

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - (b^{2} b^{1}) - b (a b)) (a^{2} + b^{2})$

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

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - b^{2 + 1} - b (a b)) (a^{2} + b^{2})$

Add $2$ and $1$ .

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - b^{3} - b (a b)) (a^{2} + b^{2})$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - b^{3} - (b \cdot b) a) (a^{2} + b^{2})$

Multiply $b$ by $b$ .

$(a^{3} + a b^{2} + a^{2} b - b a^{2} - b^{3} - b^{2} a) (a^{2} + b^{2})$

Combine the opposite terms in $a^{3} + a b^{2} + a^{2} b - b a^{2} - b^{3} - b^{2} a$ .

Reorder the factors in the terms $a^{2} b$ and $- b a^{2}$ .

$(a^{3} + a b^{2} + a^{2} b - a^{2} b - b^{3} - b^{2} a) (a^{2} + b^{2})$

Subtract $a^{2} b$ from $a^{2} b$ .

$(a^{3} + a b^{2} + 0 - b^{3} - b^{2} a) (a^{2} + b^{2})$

Add $a^{3} + a b^{2}$ and $0$ .

$(a^{3} + a b^{2} - b^{3} - b^{2} a) (a^{2} + b^{2})$

Reorder the factors in the terms $a b^{2}$ and $- b^{2} a$ .

$(a^{3} + b^{2} a - b^{3} - b^{2} a) (a^{2} + b^{2})$

Subtract $b^{2} a$ from $b^{2} a$ .

$(a^{3} - b^{3} + 0) (a^{2} + b^{2})$

Add $a^{3} - b^{3}$ and $0$ .

$(a^{3} - b^{3}) (a^{2} + b^{2})$

Step 3

Expand $(a^{3} - b^{3}) (a^{2} + b^{2})$ using the FOIL Method.

Apply the distributive property.

$a^{3} (a^{2} + b^{2}) - b^{3} (a^{2} + b^{2})$

Apply the distributive property.

$a^{3} a^{2} + a^{3} b^{2} - b^{3} (a^{2} + b^{2})$

Apply the distributive property.

$a^{3} a^{2} + a^{3} b^{2} - b^{3} a^{2} - b^{3} b^{2}$

Step 4

Simplify terms.

Simplify each term.

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

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

$a^{3 + 2} + a^{3} b^{2} - b^{3} a^{2} - b^{3} b^{2}$

Add $3$ and $2$ .

$a^{5} + a^{3} b^{2} - b^{3} a^{2} - b^{3} b^{2}$

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

Move $b^{2}$ .

$a^{5} + a^{3} b^{2} - b^{3} a^{2} - (b^{2} b^{3})$

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

$a^{5} + a^{3} b^{2} - b^{3} a^{2} - b^{2 + 3}$

Add $2$ and $3$ .

$a^{5} + a^{3} b^{2} - b^{3} a^{2} - b^{5}$

Move $b^{3}$ .

$a^{5} + a^{3} b^{2} - 1 a^{2} b^{3} - b^{5}$

Simplify (a-b)(a^2+b^2+ab)(a^2+b^2)

Related Questions