1. Graded Warm Up  Complete the graded warm up on your desk by yourself. There should be no talking.

2. 2.3 Real Zeros of Polynomial Functions

3. Back to elementary school…  Let’s say you need to find the prime factors of a rather large number. We’ll use 126 as an example. How do you start?  You pick a number that you know is a factor and divide them.  Then keep going with the answer you get from the division.  We know 2 is a factor because it’s even, so we divide 126 by 2 and get 63.  Now we can find the factors of 63, which are 9 and 7.  Now we find the factors of 9, which are 3 and 3.  We use the same idea with polynomials!

4. Long Division Reminder

5. Long Division of Polynomials  f(x) = 6x 3 – 19x 2 + 16x – 4  Where are the zeros?  We can use the zero we know to find the others.  If x=2 is a zero of the polynomial, than (x-2) is a factor.  We will divide 6x 3 – 19x 2 + 16x – 4 by (x-2) and then factor whatever is left to get the rest of the factors.

6. Back to our starting problem: finding the zeros of f(x) = 6x 3 – 19x 2 + 16x – 4  We know one of the zeros is x=2, so we can divide 6x 3 – 19x 2 + 16x – 4 by x-2  Now we’re left with 6x 2 -7x+2, which we can factor into ( )( )  So 6x 3 – 19x 2 + 16x – 4 = (x-2)(2x-1)(3x-2)  Can you find the zeros now?

7. Practice!  Divide 6x 5 + 5x 4 – 16x 3 +20x 2 -7x by 2x 2 - x

8. Practice!  Divide x 2 + 3x + 5 by x + 1  We get a remainder this time, which means that (x+1) is NOT a factor and that x=-1 is NOT a zero.  It also means that x 2 + 3x + 5 = (x+1)(x+2) + 3

9. The Division Algorithm  If f(x) and d(x) are polynomials such that d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exist unique polynomials q(x) and r(x) such that f(x) = d(x)q(x) + r(x) where r(x) = 0 or the degree of r(x) is less than the degree of d(x). Dividend Quotient Divisor Remainder

10. Synthetic Division  Only can be used for linear divisors: (x – k)  Divide by the zero, not the factor!!! 1) Divide x 4 – 10x 2 -2x + 4 by x + 3 1. Write the zero to the top left and put it in a box. Write out the coefficients of each term, remembering to insert a zero for any missing terms. 2. Bring down the first coefficient. 3. Multiply that number by the divisor. 4. Write the answer underneath the second coefficient. 5. Add and write the answer below the line. 6. Multiply this answer by the divisor 7. Lather, rinse, repeat.

11. Synthetic Practice 1. Divide -2x 3 + 3x 2 + 5x – 1 by x + 2 2. Divide x 4 - 4x 2 + 6 by x – 4

12. The Remainder Theorem  If a polynomial f(x) is divided by x – k, the remainder is r = f(k)  Use the remainder theorem to evaluate the following function at x = -2 f(x) = 3x 3 + 8x 2 + 5x – 7

13. Factor Theorem  A polynomial f(x) has a factor (x – k) if and only if f(k) = 0  You can use synthetic division to test whether a polynomial has a factor (x – k)  If the last number in your answer line is a zero, then (x – k) is a factor of f(x)

14. Repeated Division  Show that (x – 2) and (x + 3) are factors of f(x) = 2x 4 + 7x 3 – 4x 2 – 27x – 18  Then find the remaining factors of f(x)

15. Using the Remainder in Synthetic Division  In summary, the remainder r, obtained in the synthetic division of f(x) by (x – k), provides the following information.  1) The remainder r gives the value of f at x = k. That is, r = f(k)  2) If r = 0, (x – k) is a factor of f(x)  3) If r = 0, (k, 0) is an x-intercept of the graph f

16. Example (a) verify the given factors of the function (b) find the remaining factor(s) of f (c) use your results to write the complete factorization of f (d) list all real zeros of f (e) confirm your results by using a graphing utility to graph the function. f(x) = 2x 3 + x 2 – 5x + 2(x + 2) and (x – 1)

17. Rational Zero Test

18. Finding the possible real zeros 1) f(x) = x 3 + 3x 2 – 8x + 3 2) f(x) = 2x 4 – 5x 2 + 3x – 8

19. Descartes’ Rule of Signs Let f(x) = a n x n + a n-1 x n-1 + …a 2 x 2 + a 1 x + a 0 be a polynomial with real coefficients and a 0 ≠ 0 1) The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer 2) The number of negative real zeros of f is either equal to the number of variations in sign of f(-x) or less than that number by an even integer.

20. Using Descartes’ Rule of Signs Describe the possible real zeros of a) f(x) = 3x 3 – 5x 2 + 6x – 4 b) f(x)=4x 4 +3x 3 -2x 2 +7x-9

21. Finding the Zeros of a Polynomial Function f(x) = 6x 3 – 4x 2 + 3x - 2

22. Finding the Zeros of a Polynomial Function f(x) = 6x 3 – 4x 2 + 3x - 2

23. Homework Pg 123-125 #11-17 odd, 25, 27, 32, 39-43 odd