Polynomial Queries - Pushdown

For tail elements, you must "push down" lazy values before modifying individual elements.

function pushdown(block, arr, start, B)
    bStart := block * B
    for j from 0 to B - 1
        if bStart + j < n then
            arr[bStart + j] := arr[bStart + j] + start[block] + j
    start[block] := 0

Call $pushdown$ before processing tail elements in both updates and queries.

This adds $O(B) = O(\sqrt{n})$ per tail, maintaining overall $O(\sqrt{n})$ per operation.