Polynomial Queries - Analysis

For complete blocks, store two lazy values:

$1.$ $start[b]$ = the accumulated starting value for block $b$

$2.$ $count[b]$ = how many polynomial updates overlap this block

For a block starting at index $bStart$:

• Element at index $bStart + j$ gets $start[b] + count[b] \cdot j$ added

Block sum contribution from lazy: $B \cdot start[b] + count[b] \cdot \frac{B(B-1)}{2}$. This formula lets you query without pushing lazy values down.