Explanation Descartes’ Rule of Signs: Let P ( x ) be a polynomial function with real coefficients. The number of positive real zeros of P ( x ) is either equal to the number of variations in sign of P ( x ) or less than that by an even number. The number of negative real zeros of P ( x ) is either equal to the number of variations in sign of P ( − x ) or less than that by an even number. Approach for variation: If a polynomial function is written in descending powers of x and we scan it from left to right, we say that a variation in sign occurs whenever successive terms have opposite signs. Method to apply Descartes’ Rule of Signs: Step 1: We have to find the variation of the polynomial function P ( x ) . Step 2: Then by using Descartes ’ first rule of sign, we have to find the number of positive real zeros. Step 3: We have to find the variation of the polynomial function P ( − x ) . Step 4: Then by using Descartes ’ second rule of sign, we have to find the number of negative real zeros. Step 5: Finally we can find the number of non real zeros by subtracting the real zeros from the total number of zeros. Example : To find the positive, negative and non real zeros of the given function P ( x ) = 3 x 3 + 5 x 2 − 4 x + 3 . Finding positive zeros: Given, P ( x ) = 3 x 3 + 5 x 2 − 4 x + 3 The variations of the above equation are P ( x ) = 3 x 3 + 5 x 2 − ︷ + t o − 4 x + ︷ − t o + 3 Here first term and the second term has positive sign But the third term varies from + t o − . And the last term varies − t o + . So totally it has two variations in sign The number of variations in the above equation = 2 . Therefore using Descartes ’ first rule of sign, P ( x ) = 3 x 3 + 5 x 2 − 4 x + 3 has two or zero positive zeros

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ISBN: 9781305652231

Author: R. David Gustafson, Jeff Hughes

Publisher: Cengage Learning

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Chapter 4.4, Problem 62E

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Chapter 4.4, Problem 61E

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To know: The Descartes’ Rule of Signs and the method to apply it.

Solution Summary: The author explains the Descartes' Rule of Signs and the method to apply it to a polynomial function.

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To know: The Descartes’ Rule of Signs and the method to apply it.

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To know:

The Descartes’ Rule of Signs and the method to apply it.