Algebra Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Expand $(1 - a) (a - a^{2})$ using the FOIL Method.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Apply the distributive property.

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Simplify and combine like terms.

Tap for more steps...

Step 1.2.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1

Multiply $a$ by $1$ .

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

Step 1.2.1.2

Multiply $- a^{2}$ by $1$ .

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

Step 1.2.1.3

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

Tap for more steps...

Step 1.2.1.3.1

Move $a$ .

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

Step 1.2.1.3.2

Multiply $a$ by $a$ .

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

Step 1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.2.1.5

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

Tap for more steps...

Step 1.2.1.5.1

Move $a^{2}$ .

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

Step 1.2.1.5.2

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

Tap for more steps...

Step 1.2.1.5.2.1

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

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

Step 1.2.1.5.2.2

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

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

Step 1.2.1.5.3

Add $2$ and $1$ .

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

Step 1.2.1.6

Multiply $- 1$ by $- 1$ .

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

Step 1.2.1.7

Multiply $a^{3}$ by $1$ .

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

Step 1.2.2

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

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

Step 1.3

Expand $(1 + a) (1 + a^{2})$ using the FOIL Method.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify each term.

Tap for more steps...

Step 1.4.1

Multiply $1$ by $1$ .

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

Step 1.4.2

Multiply $a^{2}$ by $1$ .

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

Step 1.4.3

Multiply $a$ by $1$ .

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

Step 1.4.4

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

Tap for more steps...

Step 1.4.4.1

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

Tap for more steps...

Step 1.4.4.1.1

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

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

Step 1.4.4.1.2

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

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

Step 1.4.4.2

Add $1$ and $2$ .

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Multiply $- 2$ by $1$ .

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

Step 1.7

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

Tap for more steps...

Step 1.7.1

Move $a$ .

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

Step 1.7.2

Multiply $a$ by $a$ .

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

Step 1.8

Expand $(- 2 a - 2 a^{2}) (a - 1)$ using the FOIL Method.

Tap for more steps...

Step 1.8.1

Apply the distributive property.

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

Step 1.8.2

Apply the distributive property.

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

Step 1.8.3

Apply the distributive property.

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} - 2 a \cdot a - 2 a \cdot - 1 - 2 a^{2} a - 2 a^{2} \cdot - 1$

Step 1.9

Simplify and combine like terms.

Tap for more steps...

Step 1.9.1

Simplify each term.

Tap for more steps...

Step 1.9.1.1

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

Tap for more steps...

Step 1.9.1.1.1

Move $a$ .

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

Step 1.9.1.1.2

Multiply $a$ by $a$ .

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} - 2 a^{2} - 2 a \cdot - 1 - 2 a^{2} a - 2 a^{2} \cdot - 1$

Step 1.9.1.2

Multiply $- 1$ by $- 2$ .

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} - 2 a^{2} + 2 a - 2 a^{2} a - 2 a^{2} \cdot - 1$

Step 1.9.1.3

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

Tap for more steps...

Step 1.9.1.3.1

Move $a$ .

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

Step 1.9.1.3.2

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

Tap for more steps...

Step 1.9.1.3.2.1

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

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

Step 1.9.1.3.2.2

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

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} - 2 a^{2} + 2 a - 2 a^{1 + 2} - 2 a^{2} \cdot - 1$

Step 1.9.1.3.3

Add $1$ and $2$ .

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} - 2 a^{2} + 2 a - 2 a^{3} - 2 a^{2} \cdot - 1$

Step 1.9.1.4

Multiply $- 1$ by $- 2$ .

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

Step 1.9.2

Add $- 2 a^{2}$ and $2 a^{2}$ .

$a - 2 a^{2} + a^{3} + 1 + a^{2} + a + a^{3} + 2 a - 2 a^{3} + 0$

Step 1.9.3

Add $2 a$ and $0$ .

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

Step 2

Simplify by adding terms.

Tap for more steps...

Step 2.1

Add $a$ and $a$ .

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

Step 2.2

Add $- 2 a^{2}$ and $a^{2}$ .

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

Step 2.3

Add $a^{3}$ and $a^{3}$ .

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

Step 2.4

Combine the opposite terms in $- a^{2} + 2 a^{3} + 1 + 2 a - 2 a^{3} + 2 a$ .

Tap for more steps...

Step 2.4.1

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

$- a^{2} + 1 + 2 a + 0 + 2 a$

Step 2.4.2

Add $- a^{2} + 1 + 2 a$ and $0$ .

$- a^{2} + 1 + 2 a + 2 a$

Step 2.5

Add $2 a$ and $2 a$ .

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

Step 2.6

Move $1$ .

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