3.3 Polynomial Functions and Models

A polynomial function is a function of the form

Determine which of the following are polynomials Determine which of the following are polynomials. For those that are, state the degree. (a) Polynomial. Degree 2. (b) Not a polynomial. (c) Not a polynomial.

If f is a polynomial function and r is a real number for which f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then (a) r is an x-intercept of the graph of f. (b) (x - r) is a factor of f.

Use the above to conclude that x = -1 and x = 4 are the real roots (zeroes) of f.

1 is a zero of multiplicity 2.

. If r is a Zero or Even Multiplicity If r is a Zero or Odd Multiplicity .

Theorem If f is a polynomial function of degree n, then f has at most n-1 turning points.

Theorem For large values of x, either positive or negative, the graph of the polynomial resembles the graph of the power function.

For the polynomial (a) Find the x- and y-intercepts of the graph of f. (b) Determine whether the graph crosses or touches the x-axis at each x-intercept. (c) Find the power function that the graph of f resembles for large values of x. (d) Determine the maximum number of turning points on the graph of f.

For the polynomial (e) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis. (f) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

(b) -4 is a zero of multiplicity 1. (crosses) (a) The x-intercepts are -4, -1, and 5. y-intercept: (b) -4 is a zero of multiplicity 1. (crosses) -1 is a zero of multiplicity 2. (touches) 5 is a zero of multiplicity 1. (crosses) (d) At most 3 turning points.

Test number: -5 f (-5) 160 Graph of f: Above x-axis Point on graph: (-5, 160)

Graph of f: Below x-axis Test number: -2 f (-2) -14 Graph of f: Below x-axis Point on graph: (-2, -14)

Graph of f: Below x-axis Test number: 0 f (0) -20 Graph of f: Below x-axis Point on graph: (0, -20)

Test number: 6 f (6) 490 Graph of f: Above x-axis Point on graph: (6, 490)

(6, 490) (-1, 0) (-5, 160) (0, -20) (5, 0) (-4, 0) (-2, -14)