The sum of (𝑥 + 3) observations is (𝑥4 − 81). Find the mean of theobservations.

1. Break Down the Problem

We are given that the sum of (𝑥 + 3) observations is equal to (𝑥^4 - 81). We need to find the mean of these observations.

2. Relevant Concepts

To find the mean of a set of observations, we use the formula for the mean:

$\text{Mean} = \frac{\text{Total Sum of Observations}}{\text{Number of Observations}}$

3. Analysis and Detail

• The total sum of the observations is given as $x^4 - 81$.
• The number of observations is given as $x + 3$.

Using the mean formula, we have:

$\text{Mean} = \frac{x^4 - 81}{x + 3}$

4. Verify and Summarize

To simplify the expression, we will factor $x^4 - 81$:

$x^4 - 81 = (x^2 - 9)(x^2 + 9)$ $= (x - 3)(x + 3)(x^2 + 9)$

Now, substituting this back into the equation for the mean, we get:

$\text{Mean} = \frac{(x - 3)(x + 3)(x^2 + 9)}{x + 3}$

Assuming $x + 3 \neq 0$, we can cancel out $x + 3$:

$\text{Mean} = (x - 3)(x^2 + 9)$

Final Answer

Thus, the mean of the observations is:

$\text{Mean} = (x - 3)(x^2 + 9)$