4.2Polynomial Functions and

Models

A polynomial function is a function of the form

(a) f x x x( ) 3 4 52

Determine which of the following arepolynomials. For those that are, state thedegree.

Polynomial.

(b)

(c) Not a polynomial.

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.-3 is a zero of multiplicity 1.-5 is a zero of multiplicity 5.

.

If r is a Zero or Odd Multiplicity

If r is a Zero or Even Multiplicity

Theorem

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

TheoremFor 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) -1 is a zero of multiplicity 2. (touches) 5 is a zero of multiplicity 1. (crosses)

(d) At most 3 turning points.

(a) The x-intercepts are -4, -1, and 5. y-intercept:

Test number: -5f (-5) 160

Graph of f: Above x-axis

Point on graph: (-5, 160)

Test number: -2f (-2) -14

Graph of f: Below x-axis

Point on graph: (-2, -14)

-4 < x <-1

Test number: 0f (0) -20

Graph of f: Below x-axis

Point on graph: (0, -20)

-1 < x < 5

Test number: 6f (6) 490

Graph of f: Above x-axis

Point on graph: (6, 490)

8 6 4 2 0 2 4 6 8

300

100

100

300

500(6, 490)

(5, 0)(0, -20)

(-1, 0)

(-2, -14)(-4, 0)

(-5, 160)