1. 2.3E- Descartes Rule of Signs Descartes Rule of Signs: Used to limit the number of choices in your list of POSSIBLE rational zeros. (helpful if you don’t have a calculator!) – # of Actual Positive Solutions Count the number of SIGN CHANGES between the terms Account for imaginary solutions by subtracting multiples of 2. – # of Actual Negative Solutions Replace the X’s with (-x) & simplify each term of poly. Count the number of SIGN CHANGES between new terms Account for imaginary solutions by subtracting multiples of 2.

2. Examples: Use Descartes rule of signs to determine how many positive & negative solutions may exist. F(x) = 3x³ - 5x² + 6x – 4 – List of POSSIBLE rational zeros – # of positive zeros (sign changes): – F(-x) = – # of negative zeros (sign changes in f(-x)):

3. More examples F(x)=x³ - 2x² - 7x + 2 – List of Possible rational zeros: – # of positive zeros (sign changes): – f(-x) = – # of negative zeros (sign changes in f(-x))

4. Upper & Lower Bound Rules Rules to limit choices after you choose a number from you list that doesn’t work. Upper Bound: Answers will be smaller than it – K must be positive – Quotient in syn. ÷ are ALL POSITIVE or ZEROS Lower Bound: Answers will be larger than it – K must be negative – Quotient in syn. ÷ will ALTERNATE SIGNS & 0 = either sign.

5. Examples