Class_Notes_4.5

# Class_Notes_4.5 - Section 4.5 Class Notes The Real Zeros of...

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Section 4.5 Class Notes The Real Zeros of a Polynomial Function Example 1: Find the real zeros of the polynomial function 5 4 ( ) 4 12 3 f x x x x = + - - How many zeros could this polynomial have? Number of Real Zeros Theorem A polynomial function cannot have more real zeros than its degree. How many positive zeros could this polynomial have? Descartes’ Rule of Signs If f is a polynomial written in standard form, then the number of positive zeros either equals the number of variations in sign of the nonzero coefficients of f (x) or else equals that number less an even integer. How many negative zeros could this polynomial have? Descartes’ Rule of Signs If f is a polynomial written in standard form, then the number of negative zeros either equals the number of variations in sign of the nonzero coefficients of f(-x) or else equals that number less an even integer. What are all the possible rational zeros of this polynomial? Rational Zeros Theorem If f is a polynomial with degree 1 or higher, with integer coefficients and has the form 1 1 1 0 0 ( ) , 0, 0 n n n n n f x a x a x a x a a a - - = + + + + L then the set of fractions formed by dividing all factors of 0 a by the all factors of n a will contain all possible rational zeros of f . (Note that not all such fractions will be zeros of f . It may be that none of these fractions is a zero of

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Class_Notes_4.5 - Section 4.5 Class Notes The Real Zeros of...

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