Algebra Examples

${(a + 1)}^{7} {(a^{2} + 1)}^{10} - {(a + 1)}^{5} {(a^{2} + 1)}^{11}$

Step 1

Factor ${(a + 1)}^{5} {(a^{2} + 1)}^{10}$ out of ${(a + 1)}^{7} {(a^{2} + 1)}^{10} - {(a + 1)}^{5} {(a^{2} + 1)}^{11}$ .

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Step 1.1

Factor ${(a + 1)}^{5} {(a^{2} + 1)}^{10}$ out of ${(a + 1)}^{7} {(a^{2} + 1)}^{10}$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} ({(a + 1)}^{2}) - {(a + 1)}^{5} {(a^{2} + 1)}^{11}$

Step 1.2

Factor ${(a + 1)}^{5} {(a^{2} + 1)}^{10}$ out of $- {(a + 1)}^{5} {(a^{2} + 1)}^{11}$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} ({(a + 1)}^{2}) + {(a + 1)}^{5} {(a^{2} + 1)}^{10} (- 1 (a^{2} + 1))$

Step 1.3

Factor ${(a + 1)}^{5} {(a^{2} + 1)}^{10}$ out of ${(a + 1)}^{5} {(a^{2} + 1)}^{10} ({(a + 1)}^{2}) + {(a + 1)}^{5} {(a^{2} + 1)}^{10} (- 1 (a^{2} + 1))$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} ({(a + 1)}^{2} - 1 (a^{2} + 1))$

Step 2

Rewrite ${(a + 1)}^{2}$ as $(a + 1) (a + 1)$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} ((a + 1) (a + 1) - 1 (a^{2} + 1))$

Step 3

Expand $(a + 1) (a + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a (a + 1) + 1 (a + 1) - 1 (a^{2} + 1))$

Step 3.2

Apply the distributive property.

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a \cdot a + a \cdot 1 + 1 (a + 1) - 1 (a^{2} + 1))$

Step 3.3

Apply the distributive property.

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a \cdot a + a \cdot 1 + 1 a + 1 \cdot 1 - 1 (a^{2} + 1))$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $a$ by $a$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + a \cdot 1 + 1 a + 1 \cdot 1 - 1 (a^{2} + 1))$

Step 4.1.2

Multiply $a$ by $1$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + a + 1 a + 1 \cdot 1 - 1 (a^{2} + 1))$

Step 4.1.3

Multiply $a$ by $1$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + a + a + 1 \cdot 1 - 1 (a^{2} + 1))$

Step 4.1.4

Multiply $1$ by $1$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + a + a + 1 - 1 (a^{2} + 1))$

Step 4.2

Add $a$ and $a$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + 2 a + 1 - 1 (a^{2} + 1))$

Step 5

Apply the distributive property.

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + 2 a + 1 - 1 a^{2} - 1 \cdot 1)$

Step 6

Rewrite $- 1 a^{2}$ as $- a^{2}$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + 2 a + 1 - a^{2} - 1 \cdot 1)$

Step 7

Multiply $- 1$ by $1$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (a^{2} + 2 a + 1 - a^{2} - 1)$

Step 8

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

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (2 a + 1 + 0 - 1)$

Step 9

Add $2 a$ and $0$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (2 a + 1 - 1)$

Step 10

Subtract $1$ from $1$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (2 a + 0)$

Step 11

Factor.

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Step 11.1

Add $2 a$ and $0$ .

${(a + 1)}^{5} {(a^{2} + 1)}^{10} (2 a)$

Step 11.2

Remove unnecessary parentheses.

${(a + 1)}^{5} {(a^{2} + 1)}^{10} \cdot 2 a$