Algebra Examples

{(1 - 2 n)}^{3} - 7 n (n^{2} - 2)

${(1 - 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 1

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

a x^{2} + b x + c

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

Step 2

Simplify each term.

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Step 2.1

Use the Binomial Theorem.

1^{3} + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1^{3} + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2

Simplify each term.

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Step 2.2.1

One to any power is one.

1 + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 + 3 \cdot 1^{2} (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.2

One to any power is one.

1 + 3 \cdot 1 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 + 3 \cdot 1 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.3

Multiply

3

$3$ by

1

$1$ .

1 + 3 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 + 3 (- 2 n) + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.4

Multiply

- 2

$- 2$ by

3

$3$ .

1 - 6 n + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 3 \cdot 1 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.5

Multiply

3

$3$ by

1

$1$ .

1 - 6 n + 3 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 3 {(- 2 n)}^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.6

Apply the product rule to

- 2 n

$- 2 n$ .

1 - 6 n + 3 ({(- 2)}^{2} n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 3 ({(- 2)}^{2} n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.7

Raise

- 2

$- 2$ to the power of

2

$2$ .

1 - 6 n + 3 (4 n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 3 (4 n^{2}) + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.8

Multiply

4

$4$ by

3

$3$ .

1 - 6 n + 12 n^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 12 n^{2} + {(- 2 n)}^{3} - 7 n (n^{2} - 2)$

Step 2.2.9

Apply the product rule to

- 2 n

$- 2 n$ .

1 - 6 n + 12 n^{2} + {(- 2)}^{3} n^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 12 n^{2} + {(- 2)}^{3} n^{3} - 7 n (n^{2} - 2)$

Step 2.2.10

Raise

- 2

$- 2$ to the power of

3

$3$ .

1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)$

1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)$

1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)

$1 - 6 n + 12 n^{2} - 8 n^{3} - 7 n (n^{2} - 2)$

Step 3

Simplify the expression.

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Step 3.1

Move

1

$1$ .

- 6 n + 12 n^{2} - 8 n^{3} + 1 - 7 n (n^{2} - 2)

$- 6 n + 12 n^{2} - 8 n^{3} + 1 - 7 n (n^{2} - 2)$

Step 3.2

Move

- 6 n

$- 6 n$ .

12 n^{2} - 8 n^{3} - 6 n + 1 - 7 n (n^{2} - 2)

$12 n^{2} - 8 n^{3} - 6 n + 1 - 7 n (n^{2} - 2)$

Step 3.3

Reorder

12 n^{2}

$12 n^{2}$ and

- 8 n^{3}

$- 8 n^{3}$ .

- 8 n^{3} + 12 n^{2} - 6 n + 1 - 7 n (n^{2} - 2)

$- 8 n^{3} + 12 n^{2} - 6 n + 1 - 7 n (n^{2} - 2)$

- 8 n^{3} + 12 n^{2} - 6 n + 1 - 7 n (n^{2} - 2)

$- 8 n^{3} + 12 n^{2} - 6 n + 1 - 7 n (n^{2} - 2)$