Zeros of Polynomial Functions It can be shown that for a polynomial function of degree the following statements f n, are true. The function has, at most, real zeros. f n The graph of has, at most n–1 turning points. (Turning points, also called f relative minima or relative maxima, are points which the graph changes from increasing to decreasing or vice versa.) Real Zeros of Polynomial Functions If is a polynomial function and is a real number, the following statements are f a equivalent. is a real zero of the function x = a f. is a solution of the polynomial equation f(x) = 0 x = a (x a) is a factor of the polynomial f(x). – (a, 0) is an x-intercept of the graph of f. Example 2) Find all real zeros of . Then determine the number of turning points of the graph of the function. Repeated Zeros Definition: A factor (x a), k >1, yields a repeated zero of multiplicity k. x = a – If k is odd, the graph crosses the x-axis at x=a. If k is even, the graph touches the x-axis (but does not cross the x-axis) at x=a.
Zeros of Polynomial Functions It can be shown that for a polynomial function f of degree n, the following statements are true. The function f has, at most, n real zeros. The graph of f has, at most n–1 turning points. (Turning points, also called relative minima or relative maxima, are points which the graph changes from increasing to decreasing or vice versa.) Real Zeros of Polynomial Functions If f is a polynomial function and a is a real number, the following statements are equivalent. x = a is a real zero of the function f. x = a is a solution of the polynomial equation f(x) = 0 (x – a) is a factor of the polynomial f(x). (a, 0) is an x-intercept of the graph of f. Example 2) Find all real zeros of . Then determine the number of turning points of the graph of the function. Repeated Zeros Definition: A factor (x – a), k >1, yields a repeated zero x = a of multiplicity k. If k is odd, the graph crosses the x-axis at x=a. If k is even, the graph touches the x-axis (but does not cross the x-axis) at x=a.
Polynomial_functions_and_leading_coefficient_test_(3-3).docx