Sections 4.2 and 4.3 Zeros of Polynomial Functions. Complex Numbers

Transcription

1 Sections 4.2 and 4.3 Zeros of Polynomial Functions Complex Numbers 1

2 Sections 4.2 and 4.3 Find the Zeros of Polynomial Functions and Graph Recall from section 4.1 that the end behavior of a polynomial function is the same as the EB of the leading term ff(xx) = aa nn xx nn 1) For each polynomial function do the following: a) Give the leading term and end behavior b) Find ALL the zeros and classify them as rational-real, irrational-real, imaginary, complex. c) Identify the x-intercepts d) Write the polynomial as a product of linear factor e) Sketch a complete graph some can be done without the calculator which ones? 1. ff(xx) = (xx 2 4)(xx 2 5)(xx 3) 2. gg(xx) = (xx 2 + 4)(xx 5) 2

3 Sections 4.2 and 4.3 Zeros of Polynomial Functions Objective: Sketching complete graphs of polynomial functions (if possible) using the end behavior and the real zeros. 2) For each polynomial function do the following: a) Give the leading term and end behavior b) Find ALL the zeros and classify them as rational-real, irrational-real, imaginary, complex. c) Identify the x-intercepts d) Write the polynomial as a product of linear factor e) Sketch a complete graph some can be done without the calculator which ones? 1. h(xx) = (xx 2 2)(xx 2 2xx + 2) 2. kk(xx) = 0.5(xx 2 + 1)(xx 2 2xx + 2) 3

4 Sections 4.2 and 4.3 Zeros of Polynomial Functions Related Theorems Now we ll be dealing with polynomial functions of the form 4

5 Sections 4.2 and 4.3 Finding Zeros of Polynomial Functions Finding all zeros of a Polynomial Function when it is given on the form 3) For the polynomial function ff(xx) = 3xx 3 + 8xx 2 7xx 12. To make this process easier and faster we will not use the method outlined on the book. Follow the steps outlined below. a) How many complex zeros does the polynomial have? b) Enter the function in the Y= of the calculator c) How many real zeros (x-intercepts) does the function have? Find them by using the ZERO feature in the calculate menu. Classify them as rational-real, irrational-real, imaginary, complex. d) Write the function in factored form: 5

6 Sections 4.2 and 4.3 Zeros of Polynomial Functions Using Long Division 4) Given ff(xx) = xx 3 + 6xx 2 + 6xx 4 a) How many complex zeros does the polynomial have? b) Enter the function in the Y= of the calculator c) According to the graph, how many real zeros (x-intercepts) does the function have? Find them with the ZERO feature of the calculator. 1. Are they all rational numbers? (the listing of the p/q helps you identify the rational zeros) d) To find the irrational zeros we need to do the following: 1. Once a rational zero is identified, write the corresponding linear factor of the function. 2. Find the other factor of the function by using long division 3. Use the quadratic formula to find the zeros of any quadratic polynomial. e) Classify them as rational-real, irrational-real, imaginary, complex. f) Write the function in factored form: 6

7 Sections 4.2 and 4.3 Zeros of Polynomial Functions 5) ff(xx) = 3xx 3 7xx xx 28 a) How many complex zeros does the polynomial have? b) Enter the function in the Y= of the calculator c) According to the graph, how many real zeros (x-intercepts) does the function have? Find them with the ZERO feature of the calculator. a. Are they all rational numbers? (the listing of the p/q helps you identify the rational zeros) d) To find the irrational/complex zeros we need to do the following: a. Once a rational zero is identified, write the corresponding linear factor of the function. b. Find the other factor of the function by using long division c. Use the quadratic formula to find the zeros of any quadratic polynomial. e) Classify them as rational-real, irrational-real, imaginary, complex. f) Write the function in factored form: 7

8 Sections 4.2 and 4.3 Zeros of Polynomial Functions Conjugate Pairs Theorem 6) Use the Conjugate Pairs Theorem 7) Solve the following problems 8) Solve the following problems 9) Solve the equations: a. xx 44 xx xx = 00 b. 22xx 33 33xx = 00 Intermediate Value Theorem due next class 10) Go to section 4.2 of the book, page 212 and copy the Intermediate Value Theorem along with the graph and explanations for figure 29. Do you understand what it means? 8