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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

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Zeros of Polynomial Functions

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  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

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