polynomial help please (1 Viewer)

[bboyelement]

consider the polynomial P(x)=x^5 - ax + 1 , by considering the turning points on the curve y=P(x), prove that has 3 distinct roots if a > 5(1/2)^(8/5)

its question 4(c) of sydney boys trial 2005 if you want to see the question clearer

http://www4.tpgi.com.au/nanahcub/2005MathsExt2TrialSBHS.pdf
thanks to buchanan

 

[Yip]

P'(x)=5x^4-a=0
x=+-(a/5)^1/4
Stat points:
y1=P[(a/5)^1/4]=5^(5/4)-4a^(5/4)
y2=P[-(a/5)^1/4]=5^(5/4)+4a^(5/4)
For 3 real roots to exist, the y-coordinates of the stat points must lie on oppposite sides of the x-axis
i.e. y1y2<0
y1y2=[5^(5/4)-4a^(5/4)][5^(5/4)+4a^(5/4)]<0
5^(5/2)-16a^(5/2)<0
a>5(1/2)^8/5

 

[bboyelement]

thanks