Polynomials Summary

Algebra 2 · Axiom Academy

Let's review how polynomials extend quadratic ideas to any degree — and the theorems that let us fully factor and solve them. A polynomial's end behavior is decided entirely by two facts: the degree's parity (even/odd) and the sign of the leading coefficient — every lower-degree term becomes negligible as . Multiplicity controls how a graph meets the x -axis: odd multiplicity crosses, even multiplicity bounces — and every zero's multiplicities always sum to the degree. The division identity f(x)=d(x)q(x)+r(x) is the engine behind two shortcuts: the Remainder Theorem ( f(a) is the remainder of ) and the Factor Theorem ( (x-a) is a factor exactly when f(a)=0 ). The Rational Root Theorem narrows an infinite search to a short list of candidates — each one still has to be tested , never assumed to be a root. Descartes' Rule of Signs bounds the positive and negative real zeros by counting sign changes — but that bound can undershoot by any even number, so it is never safe to state as an exact equality. Looking ahead: the Fundamental Theorem of Algebra guarantees exactly n roots in (counted with multiplicity) for a degree- n polynomial, and when the coefficients are real, any non-real roots always arrive in conjugate pairs. As , only the leading term a_nx^n matters — every lower-degree term becomes negligible by comparison. That means end behavior is decided by just two facts: whether n is even or odd, and whether a_n is positive or negative.

This is the written version of the interactive lesson above. See the full Algebra 2 course.