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. (2 points)

Be sure to provide the answer in your explanation.

Question

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. (2 points) 

Be sure to provide the answer in your explanation.

anonymous · Answer

Zeros are the point where the graph cuts the x- axis. Actually it is the point where f(x)=0
right?

anonymous · Answer

Yes

anonymous · Answer

a good example and explanation of descartes rule of sign is found here http://www.purplemath.com/modules/drofsign.htm

anonymous · Answer

you need to count the "changes in sign" of 
$$f(x) = –3x^5 – 8x^4 +25x^3 – 8x^2 +x – 19$$

anonymous · Answer

meaning the changes in sign of the coefficients
the coefficients are $-3,-8,25,-8,1,-19$ and there are 4 changes in sign 
from -8  to 25, from 25 to -8, from -8 to 1 and from 1 to -19

anonymous · Answer

to find zeros of this.. It would be easy to factorise it completely and use table method

anonymous · Answer

this tells you that there are at most 4 positive real zeros, and you count down by 2s, meaning that there are either 4, 2 or 0 positive real zeros for this polynomial

anonymous · Answer

for the negative real zeros you check the changes in sign of $f(-x)$ and in this example $f(-x)=f(x) = 3x^5 – 8x^4 -25x^3 – 8x^2 -x – 19$

anonymous · Answer

make sure you understand the last line i wrote, because i skipped a step
replace $x$ by $-x$ and see what you get 
you should get what i wrote above
there the coefficients are $3,-8,-25,-8,-19$ and there is only one change in sign 
from 3 to -8
therefore there is at most one negative real zero, and since you count down by 2s, (and there cannot be a negative number of zeros) there MUST one one negative zero

anonymous · Answer

there are 5 zeros (counting multiplicity) in total so your options are 
1 negative zero, 4 positive
1 negative zero, 2 positive, 2 complex
1 negative zero, no positive, 4 complex

anonymous · Answer

in fact, as you can see from here, the last option is the correct one, but descartes rule of sign does not find them for you http://www.wolframalpha.com/input/?i=%E2%80%933x5+%E2%80%93+8x4+%2B25x3+%E2%80%93+8x2+%2Bx+%E2%80%93+19+

anonymous · Answer

Thanks so much