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ZEROS OF POLYNOMIAL FUNCTIONS 2.5 LINEAR FACTOR THEOREM • If f(x) is a polynomial of degree n, then f(x) can be written as a product of exactly n linear factors. • f(x) also has n complex zeros RATIONAL ZERO TEST • If a polynomial has all integer coefficients, then every rational zero of the function is of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient FIND THE RATIONAL ZEROS • 𝒇 𝒙 = 𝒙𝟑 + 𝒙 + 𝟏 FIND THE RATIONAL ZEROS • 𝒇 𝒙 = 𝟐𝒙𝟑 + 𝟑𝒙𝟐 − 𝟖𝒙 + 𝟑 WRITE THE POLYNOMIAL AS A PRODUCT OF LINEAR FACTORS GIVEN THAT 1+3𝒊 IS A ZERO • 𝒇 𝒙 = 𝒙𝟒 − 𝟑𝒙𝟑 + 𝟔𝒙𝟐 + 𝟐𝒙 − 𝟔𝟎 FIND ALL ZEROS • 𝒇 𝒙 = 𝒙𝟓 + 𝒙𝟑 + 𝟐𝒙𝟐 − 𝟏𝟐𝒙 + 𝟖 DESCARTE’S RULE OF SIGNS • Given a polynomial, the number of positive real zeros is either equal to the number of sign changes in the polynomial, or less than that by a factor of 2. • The number of negative real zeros is equal to the number of sign changes in f(-x) or or less than that by a factor of 2 DETERMINE THE NUMBER OF POSITIVE AND NEGATIVE REAL ZEROS • 𝒇 𝒙 = 𝟑𝒙𝟑 − 𝟓𝒙𝟐 + 𝟔𝒙 − 𝟒 UPPER AND LOWER BOUNDS • When looking for zeros check your synthetic division. If you divide by c and c > 0 and it does not work and all numbers on the bottom row are positive or 0, then c is an upper bound. • If c < 0 and the bottom row is alternately positive then negative (or zero), then c is a lower bound HOMEWORK • p. 164: 9-17odd, 21, 27, 37 (w/out calculator), 41 (w/out calc), 43 (w/out calc), 47, 57, 61, 65, 67, 71, 87-97odd.