Understanding Power and Polynomial Functions: Roots and Turning Points

Dec 04, 2025

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This guide covers the essential concepts of power and polynomial functions, focusing on the identification of roots and their properties. A polynomial function f(x) is defined, where r is a real number such that f(r) = 0, indicating that r is a zero or root of f and corresponds to an x-intercept on the graph. We explore the relationship between roots and factors, discussing multiplicity and its implications. Additionally, we address turning points in polynomial functions, noting that a polynomial of degree n can have at most n-1 turning points.

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Understanding Power and Polynomial Functions: Roots and Turning Points

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Power Functions and Models; Polynomial Functions and Models February 8, 2007
Power Functions f(x)=axn
Odd power
Polynomial Functions
Zero/Root of a Polynomial • If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then • r is an x-intercept of the graph of f • (x - r) is a factor of f
Multiplicity of a zero • If (x – r)m is a factor of a polynomial f and (x – r)m+1 is not a factor of f, then r is called a zero of multiplicity m of f.