Apply the Fundamental Theorem of Algebra

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Presentation transcript:

Apply the Fundamental Theorem of Algebra Notes 5.7 (Day 3) Apply the Fundamental Theorem of Algebra

Descartes’ Rule of Signs Let f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 be a polynomial function with real coefficients. The number of positive real zeros of f is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number. The number of negative real zeros of f is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

Using Descartes Rule of Signs Step 1: Make a chart with a column for positive, negative, imaginary zeros, and total zeros. Step 2: Fill in the total zeros column. (Remember, the total number of zeros is the degree of the polynomial.) Step 3: Count the number of sign changes in the given polynomial. Step 4: Record this number under the positive zeros column, as well as any other possible number of positive zeros. Step 5: Find f(-x). Step 6: Count the sign changes in f(-x). Step 7: Record this number under the negative zeros colum, as well as any other possible number of negative zeros. Step 8: Make sure your chart has all of the possible combinations listed. Remember!!!!!!! These are just the POSSIBLE numbers of zeros.

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for: f(x) = x3 + 2x – 11

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f(x) = x6 – 2x5 + 3x4 – 10x3 – 6x2 – 8x – 8.

Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for the function. g(x) = 2x4 – 8x3 + 6x2 - 3x + 1

Homework: P 384 34-41