Algebra Examples

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

Step 1

Factor $- 1$ out of $(a + b - 1) (a^{2} + 1) - a^{2} - 1$ .

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Step 1.1

Factor $- 1$ out of $(a + b - 1) (a^{2} + 1)$ .

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

Step 1.2

Factor $- 1$ out of $- a^{2}$ .

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

Step 1.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.4

Factor $- 1$ out of $- ((- a - b + 1) (a^{2} + 1)) - (a^{2})$ .

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

Step 1.5

Factor $- 1$ out of $- ((- a - b + 1) (a^{2} + 1) + a^{2}) - 1 (1)$ .

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

Step 2

Expand $(- a - b + 1) (a^{2} + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3

Simplify each term.

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Step 3.1

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

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Step 3.1.1

Move $a^{2}$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.3

Add $2$ and $1$ .

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

Step 3.2

Multiply $- 1$ by $1$ .

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

Step 3.3

Multiply $- 1$ by $1$ .

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

Step 3.4

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

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

Step 3.5

Multiply $1$ by $1$ .

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

Step 4

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

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

Step 5

Add $1$ and $1$ .

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