Question

Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = -3x^5 - 8x^4 +25x^3 - 8x^2 +x - 19 using Descartes' Rule of Signs

Answer 1

For the positive roots, we need to count the number of sign changes in the sequence formed of the polynomial's coefficients {-3,-8,+25,-8,+1,-19}. If a coefficient is zero, that term is simply omitted from the sequence. Negate the odd coefficients and do the same process to find the negative roots. In both cases, this will give you the max number of roots or less than it by an even number.

So look at the sign changes of the polynomial coefficients: {-,-,+,-,+,-|. Here, there are 4 sign changes. So there is a maximum of 4 positive coefficients. There could also be 2 or zero positive real roots.

Now change the sign of the coefficients of the odd terms: {+,-,-,-,-,-}. Here, there is only one sign change. So there is only one negative root.

In fact is there is only one real root and it is negative. The other roots are not complex:

http://www.wolframalpha.com/input/?i=roots+-3x%5E5+-+8x%5E4+%2B25x%5E3+-+8x%5E2+%2Bx+-+19

Hope this helps.

Answer 2

Thanks a ton!!

Answer 3

You're welcome!