Write in Standard Form (a+b)(a^3-3ab-b^2)

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

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

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

Step 3

Simplify each term.

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

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

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

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

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

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

Add $1$ and $3$ .

$a^{4} + a (- 3 a b) + a (- b^{2}) + b a^{3} + b (- 3 a b) + b (- b^{2})$

Rewrite using the commutative property of multiplication.

$a^{4} - 3 a (a b) + a (- b^{2}) + b a^{3} + b (- 3 a b) + b (- b^{2})$

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

Move $a$ .

$a^{4} - 3 (a \cdot a) b + a (- b^{2}) + b a^{3} + b (- 3 a b) + b (- b^{2})$

Multiply $a$ by $a$ .

$a^{4} - 3 a^{2} b + a (- b^{2}) + b a^{3} + b (- 3 a b) + b (- b^{2})$

Rewrite using the commutative property of multiplication.

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} + b (- 3 a b) + b (- b^{2})$

Rewrite using the commutative property of multiplication.

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b (a b) + b (- b^{2})$

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

Move $b$ .

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 (b \cdot b) a + b (- b^{2})$

Multiply $b$ by $b$ .

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a + b (- b^{2})$

Rewrite using the commutative property of multiplication.

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - b \cdot b^{2}$

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

Move $b^{2}$ .

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - (b^{2} b)$

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

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

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - (b^{2} b^{1})$

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

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - b^{2 + 1}$

Add $2$ and $1$ .

$a^{4} - 3 a^{2} b - a b^{2} + b a^{3} - 3 b^{2} a - b^{3}$

Step 4

Subtract $3 b^{2} a$ from $- a b^{2}$ .

Move $b^{2}$ .

$a^{4} - 3 a^{2} b - a b^{2} - 3 a b^{2} + b a^{3} - b^{3}$

Subtract $3 a b^{2}$ from $- a b^{2}$ .

$a^{4} - 3 a^{2} b - 4 a b^{2} + b a^{3} - b^{3}$

Step 5

Reorder $b$ and $a^{3}$ .

$a^{4} - 3 a^{2} b - 4 a b^{2} + a^{3} b - b^{3}$

Step 6

Move $- 3 a^{2} b$ .

$a^{4} - 4 a b^{2} + a^{3} b - b^{3} - 3 a^{2} b$

Step 7

Move $- 4 a b^{2}$ .

$a^{4} + a^{3} b - b^{3} - 4 a b^{2} - 3 a^{2} b$

Step 8

Reorder $a^{4}$ and $a^{3} b$ .

$a^{3} b + a^{4} - b^{3} - 4 a b^{2} - 3 a^{2} b$

Write in Standard Form (a+b)(a^3-3ab-b^2)

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