Mathematics Foundations/8.1 Polynomial Functions

8.1 Polynomial Functions 8.1.1 Introduction to Polynomial Functions A polynomial function is a function defined by a polynomial expression. While we studied polynomial rings in Section 7.4 from an algebraic perspective, here we focus on polynomial functions and their analytical properties. Definition A polynomial function over a field is a function of the form: where and is a non-negative integer. The integer is called the degree of the polynomial function if . 8.1.2 Basic Properties Evaluation A polynomial function can be evaluated at any value by substituting in the polynomial expression. Operations on Polynomial Functions If and are polynomial functions, then: is a polynomial function is a polynomial function (when is an integral domain) 8.1.3 Zeros of Polynomial Functions Definition A zero (or root) of a polynomial function is a value such that . Fundamental Theorem of Algebra Every non-constant polynomial function with complex coefficients has at least one complex zero. Consequence A polynomial function of degree over has exactly zeros, counting multiplicities. Multiplicity If is a factor of with multiplicity , then is a zero of with multiplicity . 8.1.4 Polynomial Division and Remainder Theorem Polynomial Division Given polynomial functions and , there exist unique polynomial functions and such that: where either or . Remainder Theorem The remainder when is divided by is equal to . Factor Theorem is a factor of if and only if . 8.1.5 Polynomial Interpolation Lagrange Interpolation Given distinct points , there exists a unique polynomial of degree at most such that for all . The Lagrange interpolation formula gives this polynomial as: Newton's Divided Difference Formula An alternative form for the interpolating polynomial is: where are the divided differences. 8.1.6 Polynomial Functions of Low Degree Constant Polynomials (Degree 0) Graph: Horizontal line Zeros: None if , all values if Linear Polynomials (Degree 1) where Graph: Straight line with slope Zero: Quadratic Polynomials (Degree 2) where Graph: Parabola Zeros: Given by the quadratic formula Discriminant: determines the number of real zeros Cubic Polynomials (Degree 3) where Graph: Cubic curve with at most two turning points Zeros: At least one real zero, with up to three real zeros 8.1.7 Behavior of Polynomial Functions End Behavior For a polynomial function with : If is even and : as If is even and : as If is odd and : as and as If is odd and : as and as Continuity and Differentiability Every polynomial function is: Continuous everywhere Differentiable everywhere Infinitely differentiable 8.1.8 Extrema of Polynomial Functions Critical Points The critical points of a polynomial function occur at values where or is undefined. Since polynomial functions are differentiable everywhere, critical points occur only when . Local Extrema A polynomial function has a local maximum or minimum at a critical point if: : Local maximum : Local minimum : Further analysis needed (higher derivatives) Number of Extrema A polynomial function of degree has at most critical points and at most local extrema. 8.1.9 Vieta's Formulas For a monic polynomial with zeros , Vieta's formulas relate the coefficients to the zeros: 8.1.10 Rational Root Theorem Theorem If is a polynomial with integer coefficients, then any rational zero (in lowest terms) satisfies: divides divides This theorem helps identify potential rational zeros of polynomials with integer coefficients. 8.1.11 Applications Polynomial functions have numerous applications in various fields: Curve fitting and data approximation Computer graphics (Bézier curves) Signal processing (filtering) Control systems Numerical analysis Physics (modeling physical phenomena) Economics (cost, revenue, and profit functions) 8.1.12 Exercises Find all zeros of the polynomial function . Use the rational root theorem to find all rational zeros of . Find a polynomial function of degree 3 with zeros at , , and . Determine the behavior of as . Find all local extrema of . Use Lagrange interpolation to find a polynomial of degree 3 that passes through the points , , , and . Prove that a polynomial function of odd degree with real coefficients must have at least one real zero.