3 3 Polynomial Functions and Models A polynomial
- Slides: 17
3. 3 Polynomial Functions and Models
A polynomial function is a function of the form
Determine which of the following are polynomials. For those that are, state the degree. (a) Polynomial. Degree 2. (b) (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. -3 is a zero of multiplicity 1. -5 is a zero of multiplicity 5.
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.
(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: f (-5) Graph of f: -5 160 Above x-axis Point on graph: (-5, 160)
-4 < x <-1 Test number: -2 f (-2) -14 Graph of f: Below x-axis Point on graph: (-2, -14)
-1 < x < 5 Test number: f (0) Graph of f: 0 -20 Below x-axis Point on graph: (0, -20)
Test number: f (6) Graph of f: 6 490 Above x-axis Point on graph: (6, 490)
(6, 490) (-5, 160) (-4, 0) (-2, -14) (-1, 0) (0, -20) (5, 0)
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- Chapter 1 graphs functions and models answers
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- How to divide a polynomial by another polynomial
- Polynomial standard form
- Chapter 4 polynomial and rational functions
- Chapter 3 polynomial and rational functions
- Chapter 6 polynomials and polynomial functions
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- Number of terms example
- Evaluate and graph polynomial functions
- Polynomial functions and their graphs
- Polynomial functions and their graphs
- Chapter 3 polynomial and rational functions
- Real world phenomena