What is the minimum number of arithmetic operations required to evaluate the…

Concept

A polynomial can be evaluated with fewer operations by factoring it into Horner form. Each multiplication and addition is counted once; powers are not computed separately when the expression is nested.

For a cubic polynomial, Horner form processes one degree at a time and avoids repeating powers of the variable.

Application

1. Set t₁ = 8z. This uses the first multiplication.

2. Set t₂ = t₁z = 8z2. This uses the second multiplication.

3. Set t₃ = t₂ + 3. This uses the first addition.

4. Set t₄ = t₃z = (8z2 + 3)z. This uses the third multiplication.

5. Set f(z) = t₄ + 12 = 8z3 + 3z + 12. This uses the second addition.

Cross-check

The nested form uses three multiplications and two additions, for a total of 5 operations. Directly forming z3, scaling both variable terms, and then adding the three terms uses four multiplications and two additions, so Horner form saves one multiplication.

Therefore, the minimum number of arithmetic operations is 5.