1 / 14

Zeros of Polynomial Functions

Zeros of Polynomial Functions. Fundamental Theorem of Algebra Conjugate Zero Theorem End behavior of a Polynomial Intermediate Value Theorem Bisection Method. Fundamental Theorem of Algebra. A complex number is said to be a zero of a polynomial function

skylar
Télécharger la présentation

Zeros of Polynomial Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Zeros of Polynomial Functions • Fundamental Theorem of Algebra • Conjugate Zero Theorem • End behavior of a Polynomial • Intermediate Value Theorem • Bisection Method KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

  2. Fundamental Theorem of Algebra A complex number is said to be a zero of a polynomial function if and only if . That is is a solution (or sometimes called a root) of the polynomial equation . A polynomial equation of degree may have real or complex roots and some of them may be repeated. Every polynomial function of degree has at least one complex zero. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  3. Repeated Zero Theorem A polynomial function of degree , has exactly complex zeros, some of which may be repeated. Furthermore, if are the distinct zeros of then where is the number of times is repeated as a zero of , and . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  4. Conjugate Zero Theorem Each zero of the polynomial function may be real or complex, and if is a non-real complex zero of then is a non-real complex zero of provided that are all real numbers. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  5. Example 1 Find the zeros and state the multiplicity and the degree of Therefore, the degree of is 9. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  6. Example 2 Find and -intercepts, determine the far behavior, state whether the graph touches or crosses the-axis The y-intercept is The x-intercepts are: As x goes to , goes to (graph raises up). And as x goes to , goes to (graph falls down). Furthermore, the graph crosses the x-axis at , and touches the x-axis at . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  7. Sketch the graph of the polynomial function Example 3 KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  8. Find a polynomial with real coefficients which satisfies the conditions Example 4 , has zeros at of multiplicity , at of multiplicity , and at of multiplicity . • But , then KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  9. Find a polynomial of least degree that has the given graph Example 5 • The polynomial has a cross at , so is a factor of . And the polynomial has a touch at so is a factor also of . • y-interceptis , • so KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  10. Intermediate Value Theorem For any polynomial , if there are two numbers and such that , then must have at least one real zero of odd multiplicity between and . Such a zero is an x-intercept of at which the graph of crosses the x-axis. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  11. Example 6 Show that (a) has at least one real zero between 1 and 2 (b) has at least two real zeros between 0 and 2. (a) Since and , it follows that has a simple zero between 1 and 2. (b) Since and , then has at least one real zero of odd multiplicity between 0 and 2. Since the number of complex zeros must be even, must have at least two real zeros. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  12. Bisection method Assume that we know that has exactly one zero of odd multiplicity within an interval , and that . Starting by the initial interval , the method approximates by constructing a sequence of smaller intervals each of which contains . We keep applying the method till the interval is small enough and we then take its midpoint as our approximation. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  13. Assume that the zero of within is unique. Example 7 • Find an interval of length which contains and then find an approximation of , estimate the error in this approximation. Then the zero is within the interval . KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

  14. Example 7 continue Now take and then the zero is within the interval . Approximate by and theis less than , half the length of the interval. KFUPM - Prep Year Math Program (c) 2009 All Right Reserved

More Related