Complex Polynomial Question Help (1 Viewer)

[kingkong123]

Hey guys i need help with this question:





What ive done so far is for part (i) I showed that the stationary points are both under the x-axis (simply by finding P'(x)=0 and subbing back into P(x)). I then proved one is a relative max and the other a relative min (by testing P''(x)). Hence there must only be one real solution with the other solutions occurring in a conjugate pair due to real co-effs. I need help finding the values which lies between.


For part (iii) i let the roots be . Then using product of roots i got but , therefore . I need help getting the values for which the modulus lies.


Also any comments on my method? Any easier method??


Thanks in advance!!

 

[Ferox]

Just sub in numbers to see when p(x) goes from negative to positive. It's between 4 and 5, which you might have guessed from part c. That means the modulus sqrt(5/a) is between sqrt(5/5) and sqrt(5/4) or between 1 and sqrt(5)/2.

 

[D94]

This question was in the PHS 2011 half yearly or trial paper

For the first part, you could use Newton's method, then by showing the root is not a whole number, you can take the whole numbers before and after. However, since the question really tells you to plug in a number because you need to know if for the following sections, I'd say the fastest way is to guess and check. It's 4 and 5, but looking at the third part, just equate beta and one of the known values in the question to determine the root is between 4 and 5.