Polynomials Summary

Pre-Calculus · Axiom Academy

Unit review: structure, end behavior, zeros and multiplicity, the division-and-theorems toolkit, and the complete-factorization theorems. A polynomial's degree and leading term set the global shape: degree n means at most n real zeros and at most n-1 turning points, and the leading term alone fixes the end behavior. A real zero c with odd multiplicity makes the graph cross the x -axis; even multiplicity makes it touch and bounce . The Remainder and Factor theorems turn "is c a zero?" into a single evaluation: (x-c) is a factor exactly when P(c)=0 . The Rational Root Theorem narrows the candidates, synthetic division tests them fast, and the Fundamental Theorem of Algebra guarantees exactly n zeros over . Over the reals, complex zeros come in conjugate pairs , so an odd-degree polynomial always has at least one real zero. Core Concept Structure & Terminology In standard form the terms are written in descending powers. The degree is the highest power n , the leading term is a_n x^n (with leading coefficient ), and the constant term a_0 is the y -intercept, since P(0)=a_0 . Degree at most n-1 turning points. Constant term a_0 = P(0) = y -intercept. Far out on either side, only the leading term matters — every lower-degree term becomes negligible. The sign of a_n and the parity of n together decide which way each "arm" of the graph points. Even degree: both arms go the same way (up if ). Odd degree: arms go opposite ways. Core Concept Zeros & Multiplicity

This is the written version of the interactive lesson above. See the full Pre-Calculus course.