Using complete sentences, describe how you would analyze the zeros of the polynomial function f(x) = –3x5 – 8x4 +25x3 – 8x2 +x – 19 using Descartes’ Rule of Signs. Be sure to provide the answer in your explanation.

Question

anonymous · Answer

possible positive real zeros, look at the number of changes in sign of the coefficents.

anonymous · Answer

$$–3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19 $$
-8 to 25   1 change
25 to -8   2 changes 
-8 to 1     3 changes 
1 to -19   4 changes

anonymous · Answer

so there are either 4, 2 or no positive roots

anonymous · Answer

now use 
$$f(-x)$$
$$f(x)=–3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19$$
$$f(-x)=-3(-x)^5-8(-x)^4+25(-x)^3-8(-x)^2-x-19$$
$$f(-x)=3x^5-8x^4-25x^3-8x^2-x-19$$

anonymous · Answer

only one change in sign, from 3 to -8
so one negative zero

anonymous · Answer

degree is 5 so there are 5 zeros all together
a) 4 positive, one negative
b) 2 positive, one negative, two complex
c) no positive, one negative, four complex

anonymous · Answer

wolfram tells me it is actually the last one that is correct, but before you know that all 3 are possible