Polynomial Functions and Their Graphs Mat 151 SLU

1. Polynomial Functions and Their Graphs Mat 151 SLU 

2. Definition of a Polynomial Function Let n be a nonnegative integer and let an, an-1,�, a2, a1, a0, be real numbers with an not 0. The function defined by f (x) = anxn + an-1xn-1 +�+ a2x2 + a1x + a0 is called a polynomial function of x of degree n. The number an, the coefficient of the variable to the highest power, is called the leading coefficient.

3. Note: Polynomial functions are often written in the factored form as: f(x) = an (x � r1 ) n1 � (x � rk )nk Where an is the leading coeffient and ri�s are the zeros of f.

5. Properties of Polynomials Domain: The set of all real numbers. Range: Depends on the degree of the polynomial. Every polynomial has a smooth continuous graph, with no holes, no corners. Every polynomial has a y-intercept. Every odd degree polynomial crosses the x-axis at least once. Every polynomial of nth degree has at most n-1 turning points.

9. Zeros of a Polynomial All the numbers for which a polynomial function f(x) = 0 are called the zeros of the polynomial function. The set of all real zeros of a polynomial function are the x-intercept of the polynomial.

11. Example Find all zeros of f (x) = 2x4 � 2. Solution: We find the zeros of f by setting f (x) equal to 0. 2x4 � 2 = 0 2(x4 � 1) = 0 2(x2 - 1)(x2 + 1) = 0 2(x - 1)(x + 1)(x2 + 1)=0 x - 1 = 0 or x + 1 = 0 or x2 + 1 = 0 X = 1 or x = -1 or x = +/-i

12. Multiplicities of Zeros If r is a zero of a polynomial function f, then we can factor f as f(x) = (x � r)k q(x) So that q(x) does not have (x-r) as a factor. Then k is the multiplicity of r.

13. Multiplicity and x-Intercepts If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.

14. Example Find the x-intercepts and multiplicity of f(x) = 2(x+2)2(x-3) Solution: x=-2 is a zero of multiplicity 2 or even x=3 is a zero of multiplicity 1 or odd

15. Graphing a Polynomial Function f (x) = anxn + an-1xn-1 + an-2xn-2 + � + a1x + a0 (an not 0) Use the Leading Coefficient Test to determine the graph's end behavior. Find x-intercepts by setting f (x) = 0 and solving the resulting polynomial equation. If there is an x-intercept at r as a result of (x - r)k in the complete factorization of f (x), then: a. If k is even, the graph touches the x-axis at r and turns around. b. If k is odd, the graph crosses the x-axis at r. c. If k > 1, the graph flattens out at (r, 0). 3. Find the y-intercept by setting x equal to 0 and computing f (0).

16. Graphing a Polynomial Function f (x) = anxn + an-1xn-1 + an-2xn-2 + � + a1x + a0 (an not 0) Use symmetry, if applicable, to help draw the graph: a. y-axis symmetry: f (-x) = f (x) b. Origin symmetry: f (-x) = - f (x). 5. Use the fact that the maximum number of turning points of the graph is n - 1 to check whether it is drawn correctly.