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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.