Precalculus Examples

$f (- \frac{5}{2}) = 20 x^{3} + 44 x^{2} - 17 x - 5$

Step 1

Replace the variable $x$ with $- \frac{5}{2}$ in the expression.

$f (- \frac{5}{2}) = 20 {(- \frac{5}{2})}^{3} + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2

Simplify the result.

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = 20 ({(- 1)}^{3} {(\frac{5}{2})}^{3}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.1.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = 20 ({(- 1)}^{3} (\frac{5^{3}}{2^{3}})) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.2

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

$f (- \frac{5}{2}) = 20 (- \frac{5^{3}}{2^{3}}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.3

Raise $5$ to the power of $3$ .

$f (- \frac{5}{2}) = 20 (- \frac{125}{2^{3}}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.4

Raise $2$ to the power of $3$ .

$f (- \frac{5}{2}) = 20 (- \frac{125}{8}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.5

Cancel the common factor of $4$ .

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Step 2.1.5.1

Move the leading negative in $- \frac{125}{8}$ into the numerator.

$f (- \frac{5}{2}) = 20 (\frac{- 125}{8}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.5.2

Factor $4$ out of $20$ .

$f (- \frac{5}{2}) = 4 (5) (\frac{- 125}{8}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.5.3

Factor $4$ out of $8$ .

$f (- \frac{5}{2}) = 4 \cdot (5 (\frac{- 125}{4 \cdot 2})) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.5.4

Cancel the common factor.

$f (- \frac{5}{2}) = 4 \cdot (5 (\frac{- 125}{4 \cdot 2})) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.5.5

Rewrite the expression.

$f (- \frac{5}{2}) = 5 (\frac{- 125}{2}) + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.6

Combine $5$ and $\frac{- 125}{2}$ .

$f (- \frac{5}{2}) = \frac{5 \cdot - 125}{2} + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.7

Multiply $5$ by $- 125$ .

$f (- \frac{5}{2}) = \frac{- 625}{2} + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.8

Move the negative in front of the fraction.

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 {(- \frac{5}{2})}^{2} - 17 (- \frac{5}{2}) - 5$

Step 2.1.9

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.9.1

Apply the product rule to $- \frac{5}{2}$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 ({(- 1)}^{2} {(\frac{5}{2})}^{2}) - 17 (- \frac{5}{2}) - 5$

Step 2.1.9.2

Apply the product rule to $\frac{5}{2}$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 ({(- 1)}^{2} (\frac{5^{2}}{2^{2}})) - 17 (- \frac{5}{2}) - 5$

Step 2.1.10

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

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 (1 (\frac{5^{2}}{2^{2}})) - 17 (- \frac{5}{2}) - 5$

Step 2.1.11

Multiply $\frac{5^{2}}{2^{2}}$ by $1$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 (\frac{5^{2}}{2^{2}}) - 17 (- \frac{5}{2}) - 5$

Step 2.1.12

Raise $5$ to the power of $2$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 (\frac{25}{2^{2}}) - 17 (- \frac{5}{2}) - 5$

Step 2.1.13

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

$f (- \frac{5}{2}) = - \frac{625}{2} + 44 (\frac{25}{4}) - 17 (- \frac{5}{2}) - 5$

Step 2.1.14

Cancel the common factor of $4$ .

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Step 2.1.14.1

Factor $4$ out of $44$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 4 (11) (\frac{25}{4}) - 17 (- \frac{5}{2}) - 5$

Step 2.1.14.2

Cancel the common factor.

$f (- \frac{5}{2}) = - \frac{625}{2} + 4 \cdot (11 (\frac{25}{4})) - 17 (- \frac{5}{2}) - 5$

Step 2.1.14.3

Rewrite the expression.

$f (- \frac{5}{2}) = - \frac{625}{2} + 11 \cdot 25 - 17 (- \frac{5}{2}) - 5$

Step 2.1.15

Multiply $11$ by $25$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 275 - 17 (- \frac{5}{2}) - 5$

Step 2.1.16

Multiply $- 17 (- \frac{5}{2})$ .

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Step 2.1.16.1

Multiply $- 1$ by $- 17$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 275 + 17 (\frac{5}{2}) - 5$

Step 2.1.16.2

Combine $17$ and $\frac{5}{2}$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 275 + \frac{17 \cdot 5}{2} - 5$

Step 2.1.16.3

Multiply $17$ by $5$ .

$f (- \frac{5}{2}) = - \frac{625}{2} + 275 + \frac{85}{2} - 5$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine the numerators over the common denominator.

$f (- \frac{5}{2}) = 275 - 5 + \frac{- 625 + 85}{2}$

Step 2.2.2

Add $- 625$ and $85$ .

$f (- \frac{5}{2}) = 275 - 5 + \frac{- 540}{2}$

Step 2.3

Find the common denominator.

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Step 2.3.1

Write $275$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = \frac{275}{1} - 5 + \frac{- 540}{2}$

Step 2.3.2

Multiply $\frac{275}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{275}{1} \cdot \frac{2}{2} - 5 + \frac{- 540}{2}$

Step 2.3.3

Multiply $\frac{275}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{275 \cdot 2}{2} - 5 + \frac{- 540}{2}$

Step 2.3.4

Write $- 5$ as a fraction with denominator $1$ .

$f (- \frac{5}{2}) = \frac{275 \cdot 2}{2} + \frac{- 5}{1} + \frac{- 540}{2}$

Step 2.3.5

Multiply $\frac{- 5}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{275 \cdot 2}{2} + \frac{- 5}{1} \cdot \frac{2}{2} + \frac{- 540}{2}$

Step 2.3.6

Multiply $\frac{- 5}{1}$ by $\frac{2}{2}$ .

$f (- \frac{5}{2}) = \frac{275 \cdot 2}{2} + \frac{- 5 \cdot 2}{2} + \frac{- 540}{2}$

Step 2.4

Combine the numerators over the common denominator.

$f (- \frac{5}{2}) = \frac{275 \cdot 2 - 5 \cdot 2 - 540}{2}$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply $275$ by $2$ .

$f (- \frac{5}{2}) = \frac{550 - 5 \cdot 2 - 540}{2}$

Step 2.5.2

Multiply $- 5$ by $2$ .

$f (- \frac{5}{2}) = \frac{550 - 10 - 540}{2}$

Step 2.6

Simplify the expression.

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Step 2.6.1

Subtract $10$ from $550$ .

$f (- \frac{5}{2}) = \frac{540 - 540}{2}$

Step 2.6.2

Subtract $540$ from $540$ .

$f (- \frac{5}{2}) = \frac{0}{2}$

Step 2.6.3

Divide $0$ by $2$ .

$f (- \frac{5}{2}) = 0$

Step 2.7

The final answer is $0$ .

$0$

Step 3