Let `f(x) = 5x^(3) + 7x^(2) + 7x + 5`, the for `x ne 0, x^(3)f(1/x)` is equal to

Options

A. `f(-x)`
B. `f(x)`
C. `1/(f(x))`
D. `1/(f(-x))`

Answer

To solve the problem, we need to find the expression for \( x^3 f\left(\frac{1}{x}\right) \) given the function \( f(x) = 5x^3 + 7x^2 + 7x + 5 \).

### Step-by-Step Solution:

1. **Substitute \( \frac{1}{x} \) into \( f(x) \)**:
We start by calculating \( f\left(\frac{1}{x}\right) \):
\[ f\left(\frac{1}{x}\right) = 5\left(\frac{1}{x}\right)^3 + 7\left(\frac{1}{x}\right)^2 + 7\left(\frac{1}{x}\right) + 5 \]
Simplifying this, we get:
\[ f\left(\frac{1}{x}\right) = 5\frac{1}{x^3} + 7\frac{1}{x^2} + 7\frac{1}{x} + 5 \]

2. **Combine the terms**:
We can rewrite \( f\left(\frac{1}{x}\right) \) as:
\[ f\left(\frac{1}{x}\right) = \frac{5}{x^3} + \frac{7}{x^2} + \frac{7}{x} + 5 \]

3. **Multiply by \( x^3 \)**:
Now we multiply \( f\left(\frac{1}{x}\right) \) by \( x^3 \):
\[ x^3 f\left(\frac{1}{x}\right) = x^3 \left( \frac{5}{x^3} + \frac{7}{x^2} + \frac{7}{x} + 5 \right) \]
Distributing \( x^3 \) to each term:
\[ x^3 f\left(\frac{1}{x}\right) = 5 + 7x + 7x^2 + 5x^3 \]

4. **Rearranging the terms**:
We can rearrange the terms:
\[ x^3 f\left(\frac{1}{x}\right) = 5x^3 + 7x^2 + 7x + 5 \]

5. **Recognizing the function**:
Notice that the expression we obtained is exactly \( f(x) \):
\[ x^3 f\left(\frac{1}{x}\right) = f(x) \]

### Final Result:
Thus, we conclude that:
\[ x^3 f\left(\frac{1}{x}\right) = f(x) \]