← Topic 1 · Algebra
AMC 10/12 · Cheatsheet

Topic 1 · Algebra

Chapter 5 · Polynomial division

📋 Reference · always available
Remainder theorem
P(x)=(x−a)Q(x)+P(a)P(x)=(x-a)Q(x)+P(a)
Factor theorem
(x−a)∣P(x)  ⟺  P(a)=0(x-a)\mid P(x) \iff P(a)=0
Rational roots
Candidate p/qp/q: p∣p\mid constant, q∣q\mid leading coeff.
Forward difference
ΔP(n)=P(n+1)−P(n)\Delta P(n) = P(n+1) - P(n)
Degree-dd poly ⇒ ΔdP\Delta^d P is constant.
Lagrange formula
P(x)=∑iyi∏j≠ix−xjxi−xjP(x) = \sum_i y_i \prod_{j\ne i}\tfrac{x-x_j}{x_i-x_j}
Newton forward
P(n+k)=∑i=0d(ki)ΔiP(n)P(n{+}k)=\sum_{i=0}^d\binom{k}{i}\Delta^i P(n)
Uniqueness
Degree ≤n\le n ⇒ uniquely determined by n+1n+1 values.