The minimum number of arithmetic operations required to evaluate the polynomial $P(X) = X^5 + 4 X^3 + 6 X + 5$ for a given value of $X$, using only one temporary variable is _______.

The polynomial can be factored as:

Using a single temporary variable $T$, the evaluation proceeds in the following sequence of arithmetic operations:

1. $T = X \times X$ (Multiplication 1: $T = X^2$)

2. $T = T + 4$ (Addition 1: $T = X^2 + 4$)

3. $T = T \times X$ (Multiplication 2: $T = X^3 + 4X$)

4. $T = T \times X$ (Multiplication 3: $T = X^4 + 4X^2$)

5. $T = T + 6$ (Addition 2: $T = X^4 + 4X^2 + 6$)

6. $T = T \times X$ (Multiplication 4: $T = X^5 + 4X^3 + 6X$)

7. $T = T + 5$ (Addition 3: $T = X^5 + 4X^3 + 6X + 5$)

This requires 4 multiplications and 3 additions, giving a total of 7 arithmetic operations.