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CHAPTER 2 POLYNOMIAL RATIONAL FUNCTIONS

2.1. COMPLEX NUMBERS. Objectives. Add

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CHAPTER 2 POLYNOMIAL RATIONAL FUNCTIONS

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    1. CHAPTER 2 POLYNOMIAL & RATIONAL FUNCTIONS

    2. 2.1 COMPLEX NUMBERS

    3. Objectives Add & subtract complex numbers Multiply complex numbers Divide complex numbers Perform operations with square roots of negative numbers

    4. i = the square root of negative 1 In the real number system, we can’t take the square root of negatives, therefore the complex number system was created. Complex numbers are of the form, a+bi, where a=real part & bi=imaginary part If b=0, a+bi = a, therefore a real number (thus reals are a subset of complex #) If a=0, a+bi=bi, therefore an imaginary # (imaginary # are a subset of complex #)

    5. Adding & Subtracting Complex # Add real to real, add imaginary to imaginary (same for subtraction) Example: (6+7i) + (3-2i) (6+3) + (7i-2i) = 9+5i When subtracting, DON’T FORGET to distribute the negative sign! (3+2i) – (5 – i) (3 – 5) + (2i – (-i)) = -2 + 3i

    6. Multiplying complex # Treat as a binomial x binomial, BUT what is i*i? It’s -1!! Why?? Let’s consider i raised to the following powers:

    7. EXAMPLE

    8. Dividing Complex # It is not standard to have a complex # in a denominator. To eliminate it, multiply be a well-chosen one: ( conjugate/conjugate) The conjugate of a+bi=a-bi We use the following fact:

    9. EXAMPLE

    10. 2.2 Quadratic Functions Objectives Recognize characteristics of parabolas Graph parabolas Determine a quadratic function’s minimum or maximum value. Solve problems involving a quadratic function’s minimum or maximum value.

    11. Quadratic functions, f(x)= graph to be a parabola. The vertex of the parabolas is at (h,k) and “a” describes the “steepness” and direction of the parabola given

    12. Minimum (or maximum) function value for a quadratic occurs at the vertex. If equation is not in standard form, you may have to complete the square to determine the point (h,k). If parabola opens up, f(x) has a min., if it opens down, f(x) has a max. This parabola opens up with a “steepness” of 2 and the minimum is at (1,1). (graph on next page)

    13. Graph of

    14. 2.3 Polynomial Functions & Their Graphs Objectives Identify polynomial functions. Recognize characteristics of graphs of polynomials. Determine end behavior. Use factoring to find zeros of polynomials. Identify zeros & their multiplicities. Use Intermediate Value Theorem. Understand relationship between degree & turning points. Graph polynomial functions.

    15. The highest degree in the polynomial is the degree of the polynomial. The leading coefficient is the coefficient of the highest degreed term. Even-degreed polynomials have both ends opening up or opening down. Odd-degreed polynomials open up on one end and down on the other end. WHY? (plug in large values for x and see!!)

    16. Zeros of polynomials When f(x) crosses the x-axis. How can you find them? Let f(x)=0 and solve. Graph f(x) and see where it crosses the x-axis. What if f(x) just touches the x-axis, doesn’t cross it, then turns back up (or down) again? This indicates f(x) did not change from pos. or neg. (or vice versa), the zero therefore exists from a square term (or some even power). We say this has a multiplicity of 2 (if squared) or 4 (if raised to the 4th power).

    17. Intermediate Value Theorem If f(x) is positive (above the x-axis) at some point and f(x) is negative (below the x-axis) at another point, f (x) = 0 (on the x-axis) at some point between those 2 pts. True for any polynomial.

    18. Turning points of a polynomial If a polynomial is of degree “n”, then it has at most n-1 turning points. Graph changes direction at a turning point.

    19. Graph

    20. Graph, state zeros & end behavior END behavior: 3rd degree equation and the leading coefficient is negative, so if x is a negative number such as -1000, f(x) would be the negative of a negative number, which is positive! (f(x) goes UP as you move to the left.) and if x is a large positive number such as 1000, f(x) would be the negative of a large positive number (f(x) goes DOWN as you move to the right.) ZEROS: x = 0, x = 3 of multiplicity 2 Graph on next page

    21. Graph f(x)

    22. Which function could possibly coincide with this graph?

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