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Consider the polynomial p(x) = a0 + a1x + a2x2 +a3x3, where ai != 0, for all i. The minimum number of multiplications needed to evaluate p on an input x is:
  • a)
    3
  • b)
    4
  • c)
    6
  • d)
    9
Correct answer is option 'A'. Can you explain this answer?
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Consider the polynomial p(x) = a0 + a1x + a2x2+a3x3, where ai != 0, fo...
Multiplications can be minimized using following order for evaluation of the given expression. p(x) = a0 + x(a1 + x(a2 + a3x))
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Consider the polynomial p(x) = a0 + a1x + a2x2+a3x3, where ai != 0, fo...
Explanation:
To evaluate the polynomial p(x) = a0 + a1x + a2x^2 + a3x^3 at an input x, we can use the following formula:
p(x) = a0 + x(a1 + x(a2 + xa3))
To evaluate p(x) using this formula, we need to perform the following multiplications:
1. x*a3
2. x*(x*a3) = x^2*a3
3. a2 + (x^2*a3)
4. x*(a2 + x^2*a3) = x*a2 + x^3*a3
5. a1 + (x*a2 + x^3*a3)
6. x*(a1 + x*a2 + x^3*a3) = x*a1 + x^2*a2 + x^4*a3
7. a0 + (x*a1 + x^2*a2 + x^4*a3)
Therefore, we need a minimum of 3 multiplications to evaluate p(x) on an input x. Hence, the correct answer is option 'A'.
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