Polynomial Functions, Zeros, Factors and Intercepts (1)

Tutorial and problems with detailled solutions on finding polynomial functions given their zeros and/or graphs and other information.

 Review Let p(x) be a polynomial function with real coefficients. If p(s) = 0 s is a zero for the polynomial function p(x). s is a solution to the equation p(x) = 0 x - s is a factor of p(x). The point (s , 0) is an x intercept of the graph of p(x). Example: let p(x) = x3 - 2·x2 + 2·x - 4 p(2) = 23 - 2*22 + 2*2 - 4 = 8 - 8 + 4 - 4 = 0 2 is a zero of p(x). x = 2 is a solution of p(x) = 0 p(x) can be written in factored form as p(x) =(x - 2)·(x2 + 2) The graph of p(x) below shows an x intercept at x = 2. TUTORIAL Example - Problem 1: The graph below is that of a polynomial function p(x) with real coefficients. The degree of p(x) is 3 and the zeros are assumed to be integers. Find p(x). Solution to Problem 1: The graph has 2 x intercepts: -1 and 2. The x intercept at -1 is of multiplicity 2. p(x) can be written as follows p(x) = a(x + 1)2(x - 2) , a is any real constant not equal to zero. To find a we need to use more information in the graph. The y intercept is at (0 , -2), which means that p(0) = -2 a(0 + 1)2(0 - 2) = -2 Solve the above equation for a to obtain a = 1 p(x) is given by p(x) = (x + 1)2(x - 2) Example - Problem 2: A polynomial function p(x) with real coefficients and of degree 5 has the zeros: -1, 2(with multiplicity 2) , 0 and 1. p(3) = -12. Find p(x). Solution to Problem 2: p(x) can be written as follows p(x) = ax(x + 1)(x - 2)2(x - 1) , a is any real constant not equal to zero. p(3) = -12 gives the following equation in a. a(3)(3 + 1)(3 - 2)2(3 - 1) = -12 Solve the above equation for a to obtain a = -1/2 p(x) is given by p(x) = -0.5x(x + 1)(x - 2)2(x - 1) The graph of p(x) is shown below. Check the intercepts and the point (3 , -12) on the graph of p(x) found above. More references and links to polynomial functions. Factor Polynomials. Polynomial Functions. Multiplicity of Zeros and Graphs Polynomials. Find Zeros of Polynomial Functions - Problems Graphs of Polynomial Functions - Self Test. Step by Step Solver to Find a Polynomial Given its Zeros and a Point. Step by Step Solver to Factor a Cubic Polynomial Given one of its Zeros. Find a Cubic Polynomial Passing Through Four Points.