Basic Math Examples

$f = - {(k^{2} - 1)}^{2} - 2 (k^{2} - 1) + 1$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(k^{2} - 1)}^{2}$ as $(k^{2} - 1) (k^{2} - 1)$ .

$f = - ((k^{2} - 1) (k^{2} - 1)) - 2 (k^{2} - 1) + 1$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f = - (k^{2} (k^{2} - 1) - 1 (k^{2} - 1)) - 2 (k^{2} - 1) + 1$

Step 1.2.2

Apply the distributive property.

$f = - (k^{2} k^{2} + k^{2} \cdot - 1 - 1 (k^{2} - 1)) - 2 (k^{2} - 1) + 1$

Step 1.2.3

Apply the distributive property.

$f = - (k^{2} k^{2} + k^{2} \cdot - 1 - 1 k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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Step 1.3.1.1.1

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

$f = - (k^{2 + 2} + k^{2} \cdot - 1 - 1 k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3.1.1.2

Add $2$ and $2$ .

$f = - (k^{4} + k^{2} \cdot - 1 - 1 k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3.1.2

Move $- 1$ to the left of $k^{2}$ .

$f = - (k^{4} - 1 \cdot k^{2} - 1 k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3.1.3

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

$f = - (k^{4} - k^{2} - 1 k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3.1.4

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

$f = - (k^{4} - k^{2} - k^{2} - 1 \cdot - 1) - 2 (k^{2} - 1) + 1$

Step 1.3.1.5

Multiply $- 1$ by $- 1$ .

$f = - (k^{4} - k^{2} - k^{2} + 1) - 2 (k^{2} - 1) + 1$

Step 1.3.2

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

$f = - (k^{4} - 2 k^{2} + 1) - 2 (k^{2} - 1) + 1$

Step 1.4

Apply the distributive property.

$f = - k^{4} - (- 2 k^{2}) - 1 \cdot 1 - 2 (k^{2} - 1) + 1$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 2$ by $- 1$ .

$f = - k^{4} + 2 k^{2} - 1 \cdot 1 - 2 (k^{2} - 1) + 1$

Step 1.5.2

Multiply $- 1$ by $1$ .

$f = - k^{4} + 2 k^{2} - 1 - 2 (k^{2} - 1) + 1$

Step 1.6

Apply the distributive property.

$f = - k^{4} + 2 k^{2} - 1 - 2 k^{2} - 2 \cdot - 1 + 1$

Step 1.7

Multiply $- 2$ by $- 1$ .

$f = - k^{4} + 2 k^{2} - 1 - 2 k^{2} + 2 + 1$

Step 2

Combine the opposite terms in $- k^{4} + 2 k^{2} - 1 - 2 k^{2} + 2 + 1$ .

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Step 2.1

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

$f = - k^{4} + 0 - 1 + 2 + 1$

Step 2.2

Add $- k^{4}$ and $0$ .

$f = - k^{4} - 1 + 2 + 1$

Step 2.3

Add $- 1$ and $1$ .

$f = - k^{4} + 0 + 2$

Step 2.4

Add $- k^{4}$ and $0$ .

$f = - k^{4} + 2$