Question

I WILL GIVE A MEDAL AND FAN WHO EVER ANSWERS THIS PLEASE HELP ME!!
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

Answer 1

Possible positive real zeros, look at the number of changes in sign of the coefficents.
–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
So there are either 4, 2 or no positive roots.
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
Only one change in sign, from 3 to -8...So one negative zero :)