Find an order for the term$-$by$-$term polynomial evaluation algorithm.

For an input array of size $n,$ the number of multiplications that are performed when the algorithm is executed equals the number of iterations of the inner loop, namely

The number of additions that are performed when the algorithm is executed equals the number of iterations of the outer loop, namely $.$

Hence, when the total number of multiplications and additions is expressed as a polynomial in $n,$ the result is

Thus, by the theorem on polynomial orders, the term$-$by$-$term polynomial algorithm has the following order.

$(n)$

$(n^2)$

$(2^n)$

$(n^3)$

$(3^n)$

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