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Transcript
7.5 Descartes' Rule, the Intermediate Value Theorem, and the Sum & Product Descartes' Rule of Signs: • The number of positive real zeros of a polynomial function, P(x), with real coefficients, is equal to the number of variations in sign of the terms of P(x) or is less than this number by a multiple of 2. • The number of negative real zeros is equal to the number of variations in sign of the terms of P(-x) or is less than this number by a multiple of 2. Example 1: Determine the possible number of positive and negative real zeros of the polynomial P(x) = x3 - 5x2 + 4x + 12. 1 Location Theorem: If P(x) is a polynomial function with real coefficients, and a and b are real numbers such that P(a) is positive and P(b) is negative, then P(x) has at least one zero between a and b. 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]. Example 2: Graph f(x) = x3 + x - 1 using a graphing calculator and locate the real zero to the nearest hundredth. 2 Sum and Product of Zeros Theroem: In the polynomial f(x) = a nxn + a n-1 xn-1 + a of the zeros is equal to -a n-1 . an The product of the zeros is equal to n-2 xn-2 + ... + a 1 x+a 0, a 0 ≠ 0, the sum a 0 for n even and -a 0 for n odd. an an Example 3: Determine the sum and product of the zeros for each of the following: a. f(x) = 4x4 - 3x3 + 2x2 - x + 6 b. g(x) = 3x5 - 2x3 + x2 - x + 4 3 Upper Bound - a number greater than or equal to the greatest zero of the function. Lower Bound - a number less than or equal to the least real zero of the function. Upper and Lower Bound Theorem: Let P(x), a polynomial function with positive leading coefficient, be divided by x-c. - If c > 0 and all the coefficients in the quotient and remainder are nonnegative, then c is an upper bound of the zeros. - If c < 0 and the coefficients in the quotient and remainder alternate in sign, then c is a lower bound of the zeros. Example 4: Determine the least integral upper bound and the greates integral lower bound for f(x) = x5 + 5x4 - 3x3 - 29x2 + 2x + 23. 4 Example 5: Sketch a graph of P(x) that satisfies the following conditions: • P(x) is 4th degree with leading coefficient 1 • P(x) has two distinct negative real zeros and one positive real zero with multiplicity 2. • The greatest integral lower bound for the zeros of P(x) is -5. • The least integral upper bound is 4. 5