Real Zeros of Polynomials Name:____________________ Worked With:_________________________________ Instructions: It is expected that you have watched the section video, read the book, or at least have your book available for reference. If you are working in a group, take a few minutes to introduce yourselves. Please answer each question as completely as possible. Have fun! When modeling real-life situations, the x-intercepts of a function often have an important meaning. For example, suppose the height of this motorcyclist is modeled by ht =−20t 2  40t where t is the amount of time in seconds since he started his jump. Then solving ht = 0 would show at what time he left the ground and when exactly he landed. 1. Find the x-intercepts of the function discussed above by solving ht =−20t 2  40t . Then interpret your answers. Do they seem reasonable? In keeping with our tradition of renaming concepts (remember when slope became Average Rate of Change?), we will start calling x-intercepts the "real zeros" of a function. It may have taken you a few minutes to find the real zeros. You probably either graphed it and found the xintercepts or used the Quadratic Formula . A few of you may even have factored it as −20t t − 2 = 0 and realized that solving 20 t = 0 and t−2 = 0 would give the two answers. Since the Quadratic Formula only works with quadratics, graphing and factoring are the only two methods we can apply to polynomials in general. 2. Find the real zeros of p x  =  x  2 x − 3 x  1 x − 4 without graphing. Then use your calculator to double check your answers. 3. Discuss how you could use a graph to factor p x  =x 3  5 x 2 − 2 x − 24 . Created by Nolan Mitchell at Chemeketa Community College Photo from Jean-Marc Favre's Flickr site It is important to recognize, as you saw in #2, that if  x  2 is a factor of a polynomial, then it has a real zero (or x-intercept) at x =−2 . It's also true, as you found in #3, that if it has an x-intercept at 3 then  x−3 is one of its factors. 4. Find the factored form of polynomial with real zeros at x=1 , x=2 and x=−3 . 5. Look closely at the graph of p x  =  x−2 x1 x1 =  x−2 x1 2 . What unusual thing happens at x =−1 that does not happen at x = 2 ? 6. Graph p  x  =  x−2 x1 x1 x1 =  x−2 x1 3 ? What happens at x=−1 now? 7. Go back to #6 and try different powers for the  x1 term. Discuss what you see. 8. Write the completely factored form for the following polynomials. Polynomial Completely Factored Form A 5th degree polynomial with multiple zeros at x = 1 and x =−3 . A 4th degree polynomial with multiple zeros at x = 2 and x =−5 . -4 -3 -2 -1 1 2 3 4 -8 -6 -4 -2 2 4 6 8 x y -4 -3 -2 -1 1 2 3 4 -8 -6 -4 -2 2 4 6 8 x y Created by Nolan Mitchell at Chemeketa Community College

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