Polynomial Functions Polynomials A Polynomial in one variable, x, is an expression of the form 0 n 1 2 an x a1 x ... an 2 x an 1 x an The coefficients represent complex numbers (real or imaginary), ao is not zero, and n represents a nonnegative integer. The degree of a polynomial is a greatest exponent of the variable. The leading coefficient is a coefficient of the variable with the greatest exponent. Zeros of a polynomial function are values of x for which f(x) = 0 Example 1 Consider the polynomial function f(x) = x3 - x2 - 7x + 3. a. State the degree and leading coefficient of the polynomial. x3 - x2 - 7x + 3 has a degree of 3 and a leading coefficient of 1. b. Determine whether 3 is a zero of f(x). Evaluate f(x) = x3 - x2 - 7x + 3 for x = 3. That is, find f(3). f(3) = (3)3 - (3)2 - 7(3) + 3 x = 3 f(3) = 27 - 9 - 21 + 3 f(3) = 0 Since f(3) = 0, 3 is a zero of f(x) = x3 - x2 - 7x + 3. Fundamental Thm. Of Algebra Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers. COROLLARY: A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers. P (x) = k(x -r1)(x - r2)(x - r3) …(x - rn ) Example 2 a. Write a polynomial equation of least degree with roots 3, 2i, and -2i. b. Does the equation have an odd or even degree? How many times does the graph of the related function cross the x-axis? a. If x = 3, then x - 3 is a factor of the polynomial. Likewise, if x = 2i and x = -2i, then x - 2i and x - (-2i) are factors of the polynomial. Therefore, the linear factors for the polynomial are x - 3, x - 2i, and x + 2i. Now find the products of these factors. (x - 3)(x - 2i)(x + 2i) = 0 (x - 3)(x2 - 4i2) = 0 (x - 3)(x2 + 4) = 0 -4i2 = -4(-1) or 4 x3 - 3x2 + 4x - 12 = 0 A polynomial with roots 3, 2i, and -2i is x3 - 3x2 + 4x - 12 = 0. b. The degree of this equation is 3. Thus, the equation has an odd degree since 3 is an odd number. Since two of the roots are imaginary, the graph will only cross the x-axis once. Example 3 State the number of complex roots of the equation 4x4 - 3x2 - 1 = 0. Then find the roots and graph the related function. Factor the equation to find the roots. 4x4 - 3x2 - 1 = 0 (4x2 + 1)(x2 - 1) = 0 (4x2 + 1)(x + 1)(x - 1) = 0 To find each root, set each factor equal to zero. (4x2 + 1)(x + 1)(x - 1) = 0 2 4x + 1= 0 1 x2 = - 4 1 x = 4 1 x = i 2 x + 1= 0 x - 1 = 0 x= -1 x = 1 The roots are , -1, and 1..