7.5 Descartes’ Rule, Intermediate Value Theorem, Sum & Product of Zeros

1. 7.5 Descartes’ Rule, Intermediate Value Theorem, Sum & Product of Zeros

2. We are going to learn more rules to help us better factor and find zeros of polynomial functions! • Descartes’ Rule of Signs • Determines the possible nature of the zeros  # of (+) or (–) real roots as well as imaginary roots How? • Count # of sign changes in polynomial this is # of (+) real roots or less by a multiple of 2 • Plug in (–x) for x & count # of sign changes again this is # of (–) real roots or less by a multiple of 2 • Enter in a table (+) (–) i *remembering each row must add to degree Ex 1) Determine the nature of zeros (+) (–) i –  2 + + + 2 1 0 – – 0 1 2 –  1 + P(–x) =

3. Intermediate Value Theorem • If P(x) is a polynomial function such that a < b and P(a) ≠ P(b), then P(x) takes on every value between P(a) and P(b) in the interval [a, b] *Special Case: If the values of P(x) change from (+) to (–) or from (–) to (+), there must be a 0 somewhere in between Ex 2) Use graphing calculator to find zeros to nearest hundredth 0.45, –4.03 • Upper & Lower Bound Theorems: Using synthetic division on poly with real coeff & lead coeff is positive: • (1) If c is a (+) real, & all #s in last row are non-negative, c is an • upper bound • (2) If c is a (–) real, & the #s in last row alternate in sign, c is a • lower bound • (0 can be written as +0 or –0) • *Start at 0 & go up for upper bound, go down for lower bound (no zeros > c) (no zeros < c)

4. Ex 3) Determine the least integral upper bound and greatest integral lower bound for 2 –1 2 –1 –24 12 1 2 1 3 2 –22 Stop! negative 2 2 3 8 15 6 24  all (+) : upper bound –1 –18 2 –3 5 –6 Stop! no switch –2 2 –5 12 –25 26 –40  alternates : lower bound

5. Ex 4) Sketch a possible graph P(x) satisfying the following conditions: • 4th degree, leading coefficient 1 • 2 distinct negative real zeros & a positive real zero with mult 2 • greatest integral lower bound is –5 and least integral upper bound is 4 (edges going up) • (crosses x-axis at 2 (–) values & is tangent at a (+) value) (zeros happen between –5 and 4) (Do on board) • –5 4 (just one possible answer)

6. Putting it all together! Ex 5) Locate the real zeros of the polynomial between consecutive integers (+) (–) i –  1 + – – nature of roots? 1 3 0 + + –  3 – P(–x) = 1 1 2 *try for (+) first 1 0 –3 –6 –2 1 1 1 –2 –8 –10 –10 2 1 2 1 –4 here! 3 is upper bound! 3 1 3 6 12 34 0 1 0 –3 –6 –2 here! more? Between 2 & 3 and –1 & 0 –1 1 –1 –2 –4 2 lower bound so no more! –2 1 –2 1 –8 14

7. Homework #705 Pg 361 #1–15 odd, 16, 20–24, 28, 30, 36, 40, 42, 45, 47, 48, 51 Hint: For #48, read paragraph above