1.What is the upper bound of the function?
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upper bound =
wouldnt than answer be just 13 since when you do long divsion 13/4 -25 -5 -13 it is 4 27 346 and 4485 which are all postitive ?
Divide each term of the polynomial by 4
4x^4 - 25x^2 - 5x - 13
___ _____ ___ ___ = (1x^4) - ( 25/4)x^2 - (5/4)x - 13/4
4 4 4 4
Write down the coefficients = 1 - 25/4 -5/4 -13/4
Drop the first one and remove the minus signs from any of the rest ....we get ... 25/4 5/4 13/4
We have two possible bounds on the polynomial
a) Take the largest value of the above and add 1 = 25/4 + 1 = 25/4 + 4/4 = 29/4 = 7.25
b) Take the larger of : the sum of the coefficients or 1....Add the coefficients = 43/4.....this is greater than 1....so the second possible bound is 43/4 = 10.75
Select the smaller of (a) or (b) = 7.25
This tells us that the upper and lower bounds on zeroes are 7.25 and - 7.25
So.....the upper bound is x = 7.25.....in other words......no real root will be > 7.25
Is it possible for it to be 7 ? Because when I didive 7 by the polyonmial I get all postitive values and that is lower than 7.25?
I don't quite understand what you mean.......we are looking for the possible upper bound on zeroes.....in other words.....we are trying to find the largest x value such that no positive root can exceed this upper bound
Look at the graph here : https://www.desmos.com/calculator/omunuy8c4t
Note that the largest real root occurs at about (2.68, 0).....so this confirms the fact that we do not need to search for any real roots > 7.25 because they do not exist
If you click on this link the upper boun therm states "by using synthetic divsion if alll numbers are postitive and then thats the upper bound "
When i solve 7 into the equation so 7/4 -25 -5 -13 , i got all postitive values and then once i do 6 it is a mis of pos and neg values