Precalculus Examples

$p (t) = \frac{(t^{2} - 4) (t + 1)}{t^{2} + 3}$ on $[- 3, 1]$

Step 1

Write $p (t) = \frac{(t^{2} - 4) (t + 1)}{t^{2} + 3}$ as an equation.

$y = \frac{(t^{2} - 4) (t + 1)}{t^{2} + 3}$

Step 2

Substitute using the average rate of change formula.

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Step 2.1

The average rate of change of a function can be found by calculating the change in $y$ values of the two points divided by the change in $t$ values of the two points.

$\frac{f (1) - f (- 3)}{(1) - (- 3)}$

Step 2.2

Substitute the equation $y = \frac{(t^{2} - 4) (t + 1)}{t^{2} + 3}$ for $f (1)$ and $f (- 3)$ , replacing $t$ in the function with the corresponding $t$ value.

$\frac{(\frac{({(1)}^{2} - 4) ((1) + 1)}{{(1)}^{2} + 3}) - (\frac{({(- 3)}^{2} - 4) ((- 3) + 1)}{{(- 3)}^{2} + 3})}{(1) - (- 3)}$

Step 3

Simplify the expression.

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Step 3.1

Multiply the numerator and denominator of the fraction by $(1^{2} + 3) ({(- 3)}^{2} + 3)$ .

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Step 3.1.1

Multiply $\frac{\frac{(1^{2} - 4) (1 + 1)}{1^{2} + 3} - \frac{({(- 3)}^{2} - 4) (- 3 + 1)}{{(- 3)}^{2} + 3}}{1 - (- 3)}$ by $\frac{(1^{2} + 3) ({(- 3)}^{2} + 3)}{(1^{2} + 3) ({(- 3)}^{2} + 3)}$ .

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

Step 3.1.2

Combine.

$\frac{(1^{2} + 3) ({(- 3)}^{2} + 3) (\frac{(1^{2} - 4) (1 + 1)}{1^{2} + 3} - \frac{({(- 3)}^{2} - 4) (- 3 + 1)}{{(- 3)}^{2} + 3})}{(1^{2} + 3) ({(- 3)}^{2} + 3) (1 - (- 3))}$

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify by cancelling.

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Step 3.3.1

Cancel the common factor of $1^{2} + 3$ .

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Step 3.3.1.1

Cancel the common factor.

Step 3.3.1.2

Rewrite the expression.

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

Step 3.3.2

Cancel the common factor of ${(- 3)}^{2} + 3$ .

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Step 3.3.2.1

Move the leading negative in $- \frac{({(- 3)}^{2} - 4) (- 3 + 1)}{{(- 3)}^{2} + 3}$ into the numerator.

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

Step 3.3.2.2

Factor ${(- 3)}^{2} + 3$ out of $(1^{2} + 3) ({(- 3)}^{2} + 3)$ .

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

Step 3.3.2.3

Cancel the common factor.

Step 3.3.2.4

Rewrite the expression.

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

Step 3.4

Simplify the numerator.

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Step 3.4.1

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

$\frac{(9 + 3) (1^{2} - 4) (1 + 1) + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.2

Add $9$ and $3$ .

$\frac{12 (1^{2} - 4) (1 + 1) + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.3

One to any power is one.

$\frac{12 (1 - 4) (1 + 1) + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.4

Subtract $4$ from $1$ .

$\frac{12 \cdot - 3 (1 + 1) + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.5

Multiply $12$ by $- 3$ .

$\frac{- 36 (1 + 1) + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.6

Add $1$ and $1$ .

$\frac{- 36 \cdot 2 + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.7

Multiply $- 36$ by $2$ .

$\frac{- 72 + (1^{2} + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.8

One to any power is one.

$\frac{- 72 + (1 + 3) (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.9

Add $1$ and $3$ .

$\frac{- 72 + 4 (- ({(- 3)}^{2} - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.10

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

$\frac{- 72 + 4 (- (9 - 4) (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.11

Subtract $4$ from $9$ .

$\frac{- 72 + 4 (- 1 \cdot 5 (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.12

Multiply $- 1$ by $5$ .

$\frac{- 72 + 4 (- 5 (- 3 + 1))}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.13

Add $- 3$ and $1$ .

$\frac{- 72 + 4 (- 5 \cdot - 2)}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.14

Multiply $4 (- 5 \cdot - 2)$ .

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Step 3.4.14.1

Multiply $- 5$ by $- 2$ .

$\frac{- 72 + 4 \cdot 10}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.14.2

Multiply $4$ by $10$ .

$\frac{- 72 + 40}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.4.15

Add $- 72$ and $40$ .

$\frac{- 32}{(1^{2} + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5

Simplify the denominator.

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Step 3.5.1

One to any power is one.

$\frac{- 32}{(1 + 3) ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.2

Add $1$ and $3$ .

$\frac{- 32}{4 ({(- 3)}^{2} + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.3

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

$\frac{- 32}{4 (9 + 3) \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.4

Add $9$ and $3$ .

$\frac{- 32}{4 \cdot 12 \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.5

Multiply $4 \cdot 12 \cdot 1$ .

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Step 3.5.5.1

Multiply $4$ by $12$ .

$\frac{- 32}{48 \cdot 1 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.5.2

Multiply $48$ by $1$ .

$\frac{- 32}{48 + (1^{2} + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.6

One to any power is one.

$\frac{- 32}{48 + (1 + 3) ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.7

Add $1$ and $3$ .

$\frac{- 32}{48 + 4 ({(- 3)}^{2} + 3) (- (- 3))}$

Step 3.5.8

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

$\frac{- 32}{48 + 4 (9 + 3) (- (- 3))}$

Step 3.5.9

Add $9$ and $3$ .

$\frac{- 32}{48 + 4 \cdot 12 (- (- 3))}$

Step 3.5.10

Multiply $4 \cdot 12 (- (- 3))$ .

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Step 3.5.10.1

Multiply $4$ by $12$ .

$\frac{- 32}{48 + 48 (- (- 3))}$

Step 3.5.10.2

Multiply $- 1$ by $- 3$ .

$\frac{- 32}{48 + 48 \cdot 3}$

Step 3.5.10.3

Multiply $48$ by $3$ .

$\frac{- 32}{48 + 144}$

Step 3.5.11

Add $48$ and $144$ .

$\frac{- 32}{192}$

Step 3.6

Reduce the expression by cancelling the common factors.

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Step 3.6.1

Cancel the common factor of $- 32$ and $192$ .

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Step 3.6.1.1

Factor $32$ out of $- 32$ .

$\frac{32 (- 1)}{192}$

Step 3.6.1.2

Cancel the common factors.

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Step 3.6.1.2.1

Factor $32$ out of $192$ .

$\frac{32 \cdot - 1}{32 \cdot 6}$

Step 3.6.1.2.2

Cancel the common factor.

$\frac{32 \cdot - 1}{32 \cdot 6}$

Step 3.6.1.2.3

Rewrite the expression.

$\frac{- 1}{6}$

Step 3.6.2

Move the negative in front of the fraction.

$- \frac{1}{6}$