Question

The minimum number of arithmetic operations required to evaluate the polynomial

P(X) = X⁵ + 4X³ + 6X + 5 for a given value of X, using only on temporary variable is _______

Answer 1

Concept:

To find the number of operations using a temporary variable, we need to simplify the equation to obtain bracket pairs and arithmetic signs.

Explanation:

P(X) = X⁵ + 4X³ + 6X + 5

P(X) = X (X⁴ + 4X² + 6) + 5

P(X) = X (X (X³ + 4X) + 6) + 5

P(X) = X (X (X (X² + 4)) + 6) + 5

P(X) = X (X (X (X (X) + 4)) + 6) + 5

Method 1 -

Let T be a temporary variable to store intermediate results.

1. T = (X) * (X)
2. T = T + 4
3. T = (X) * (T)
4. T = (X) * (T)
5. T = T + 6
6. T = (X) * T
7. T = T + 5

 

Hence, 7 operations are used with one temporary variable.

Method 2 -

Counting number of bracket pairs gives the number of multiplication operations = 4

Counting number of + signs gives the number of addition operations = 3

Total operations = 7