Algebra Examples

${(x^{2} - 1)}^{4}$

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(x^{2})}^{4 - k} \cdot {(- 1)}^{k}$

Step 2

Expand the summation.

$\frac{4!}{(4 - 0)! 0!} \cdot {(x^{2})}^{4 - 0} \cdot {(- 1)}^{0} + \frac{4!}{(4 - 1)! 1!} \cdot {(x^{2})}^{4 - 1} \cdot {(- 1)}^{1} + \frac{4!}{(4 - 2)! 2!} \cdot {(x^{2})}^{4 - 2} \cdot {(- 1)}^{2} + \frac{4!}{(4 - 3)! 3!} \cdot {(x^{2})}^{4 - 3} \cdot {(- 1)}^{3} + \frac{4!}{(4 - 4)! 4!} \cdot {(x^{2})}^{4 - 4} \cdot {(- 1)}^{4}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(x^{2})}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x^{2})}^{4}$ by $1$ .

${(x^{2})}^{4} \cdot {(- 1)}^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.2

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2 \cdot 4} \cdot {(- 1)}^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.2.2

Multiply $2$ by $4$ .

$x^{8} \cdot {(- 1)}^{0} + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.3

Anything raised to $0$ is $1$ .

$x^{8} \cdot 1 + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.4

Multiply $x^{8}$ by $1$ .

$x^{8} + 4 \cdot {(x^{2})}^{3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.5

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 4.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} + 4 \cdot x^{2 \cdot 3} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.5.2

Multiply $2$ by $3$ .

$x^{8} + 4 \cdot x^{6} \cdot {(- 1)}^{1} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.6

Evaluate the exponent.

$x^{8} + 4 x^{6} \cdot - 1 + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.7

Multiply $- 1$ by $4$ .

$x^{8} - 4 x^{6} + 6 \cdot {(x^{2})}^{2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.8

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 4.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} - 4 x^{6} + 6 \cdot x^{2 \cdot 2} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.8.2

Multiply $2$ by $2$ .

$x^{8} - 4 x^{6} + 6 \cdot x^{4} \cdot {(- 1)}^{2} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.9

Raise $- 1$ to the power of $2$ .

$x^{8} - 4 x^{6} + 6 x^{4} \cdot 1 + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.10

Multiply $6$ by $1$ .

$x^{8} - 4 x^{6} + 6 x^{4} + 4 \cdot {(x^{2})}^{1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.11

Multiply the exponents in ${(x^{2})}^{1}$ .

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Step 4.11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} - 4 x^{6} + 6 x^{4} + 4 \cdot x^{2 \cdot 1} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.11.2

Multiply $2$ by $1$ .

$x^{8} - 4 x^{6} + 6 x^{4} + 4 \cdot x^{2} \cdot {(- 1)}^{3} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.12

Raise $- 1$ to the power of $3$ .

$x^{8} - 4 x^{6} + 6 x^{4} + 4 x^{2} \cdot - 1 + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.13

Multiply $- 1$ by $4$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + 1 \cdot {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.14

Multiply ${(x^{2})}^{0}$ by $1$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + {(x^{2})}^{0} \cdot {(- 1)}^{4}$

Step 4.15

Multiply the exponents in ${(x^{2})}^{0}$ .

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Step 4.15.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + x^{2 \cdot 0} \cdot {(- 1)}^{4}$

Step 4.15.2

Multiply $2$ by $0$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + x^{0} \cdot {(- 1)}^{4}$

Step 4.16

Anything raised to $0$ is $1$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + 1 \cdot {(- 1)}^{4}$

Step 4.17

Multiply ${(- 1)}^{4}$ by $1$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + {(- 1)}^{4}$

Step 4.18

Raise $- 1$ to the power of $4$ .

$x^{8} - 4 x^{6} + 6 x^{4} - 4 x^{2} + 1$