1. Homework “Mini-Quiz” Use the paper provided - 10 min. (NO TALKING!!) Do NOT write the question – Answer Only!! 1)A polynomial function written with terms in descending degree is written in _______ form. 2)In the function above, n is called ______________. 3)Describe how f(x) = -(x+2) 3 - 1 would change g(x) = x 3. 4)Given any polynomial function f(x) = a n x n +..+a 1 x+a 0, if a n > 0 and n is ____, then and 5)Given any polynomial function f(x) = a n x n +..+a 1 x+a 0, if a n ___ 0 and n is even, thenand 6)Graph f(x) = (x-2) 2 (x+1)(x-3). Describe the end behavior using limits. (#21) 7)Find the zeros of f(x) = x 3 -25x algebraically. (Show your work) (#36) If you finish before the timer sounds, start the warm-up. (NO TALKING!!)

2. Warm-up (5 min.) 1)Using only algebra, find a cubic function with the given zeros: 3,-4,6 2)Use cubic regression to fit a curve through the four points given in the table: x-2 1 4 7 y 2 5 9 26

3. 2.4 Real Zeros of Polynomial Functions Divide polynomials using long division or synthetic division Apply the Remainder and Factor Theorem Find upper and lower bounds for zeros of polynomials

4. Do you recall? In the long division shown, what are the names for the values: 30, 4, 7, and 2?

5. Ex1 Using long division to divide f(x) by d(x), and write a summary statement in polynomial form and fraction form. a)f(x) = x 2 - 2x + 3; d(x) = x – 1 b)f(x) = x 4 - 2x 3 + 3x 2 - 4x + 6; d(x) = x 2 + 2x - 1

6. Long Division and the Division Algorithm Let f(x) and d(x) be polynomials with the degree of f greater than or equal to the degree of d, and d(x)  0. Then there are unique polynomials q(x) and r(x), called the quotient and remainder, such that f(x) = d(x) q(x) + r(x) where either r(x) = 0 or the degree of r is less than the degree of d.

7. The remainder determines a factor Remainder Theorem - If a polynomial f(x) is divided by x - k, then the remainder is r = f(k). Factor Theorem - A polynomial function f(x) has a factor x - k if an only if f(k) = 0.

8. Ex 2 Use the remainder theorem to find the remainder when f(x) = 2x 2 - 3x + 1 is divided by: a)x – 2 b)x + 4

9. Fundamental Connections for Polynomial Functions For a polynomial function f and a real number k, the following statements are equivalent: 1.x = k is a solution (or root) of the equation f(x) = 0. 2.k is a zero of the function f. 3.k is an x-intercept of the graph of y = f(x). 4.x - k is a factor of f(x).

10. Ex 3 Use synthetic division to divide f(x) = x 3 + 5x 2 + 3x – 2 by: a)x + 1 b)x - 2

11. Synthetic Division 1.Express the polynomial in standard form. 2.Use the coefficient (including zero coefficients) for synthetic division. 3.Find the zero of the divisor (x - k = 0) 4.Use this as the divisor in the synthetic division 5.Bring down the leading coefficient 6.Multiply by the “zero divisor” 7.Add this product to the next coefficient 8.Repeat steps 4 & 5 until all coefficients have been used 9.The last coefficient is the remainder 10.The other coefficient are the coefficients for the quotient polynomial when written in standard form.

12. Upper and Lower Bounds Let f be a polynomial function of degree n > 1 with a positive leading coefficient. Suppose f(x) is divided by x – k using synthetic division. If k > 0 and every number in the last line is nonnegative (positive or zero), then k is an upper bound for the real zeros of f. If k < 0 and the numbers in the last line are alternately nonnegative and nonpositive, then k is a lower bound for the real zeros of f.

13. Ex 4 Use synthetic division to prove that the number k is the upper or lower bound (as stated) for the real zeros of the function f. a)k = 3 is an upper bound; f(x) = 2x 3 – 4x 2 + x – 2 b)k = -1 is a lower bound; f(x) = 3x 3 – 4x 2 + x + 3

14. Searching for zeros Ex 5 Show that all the zeros of f(x) = 2x 3 – 3x 2 – 4x + 6 lie within the interval [-7,7]. Find all of the zeros.

15. Rational Zeros (Roots) Theorem If a polynomial f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 has any rational roots, then they are of the form p/q where q is a factor of a n (the leading coefficient) and p is a factor of a 0 (the constant term). Ex 6 List all the possible rational roots of f(x) = 2x 3 + 5x 2 - 3x + 5

16. Tonight’s Assignment p. 216 - 218 Ex 3-33 m. of 3, 39, 42, 51-60 m. of 3

17. Exit Ticket Find all the roots of f(x) from Ex 6 and place in the turn in box before you leave. Have a great day!! Remember to study!!