Question

Consider the polynomial function.

Answer 1

The number of positive zeros is 1. The number of negative zeros is either 3 or 1.

Rule of Descartes

This states that the number of real positive zeros of a polynomial are equal to or less than by an even number the number of sign changes of the coefficients of the polynomial, f(x). Also, the number of real negative zeros of a polynomial are equal to or less than by an even number the number of sign changes of the coefficients of the polynomial, f(-x).

The positive zero

Since f(x) = x⁴ + 2x³ - 11x² - 5x - 6

The cofficients are +1, + 2, -11, -5, -6

There is no sign change from + 1 to + 2.

There is a sign change from + 2 to - 11.

There is no sign change from - 11 to - 5.

There is no sign change from -5 to - 6.

Since there is only one sign change, there is 1 positive zero.

The negative zero

f(-x) = x⁴ - 2x³ - 11x² + 5x - 6

The cofficients are +1, - 2, -11, +5, -6

There is a sign change from + 1 to - 2.

There is no sign change from - 2 to - 11.

There is a sign change from - 11 to + 5.

There is a sign change from +5 to - 6.

Since there are 3 sign changes, we have 3 or 3 - 2 = 1 negative zeros.

So, the number of positive zeros is 1. The number of negative zeros is either 3 or 1.

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