3okie sir.
p, q, r are the roots of ax^3+bx^2+cx+d=0 find the cubic equation whose roots are p^3+\frac{b}{a}\times p^2, q^3+\frac{b}{a}\times q^2 \text{ and } r^3+\frac{b}{a}\times r^2
This one is from FIITJEE AITS..
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26 Answers
1sir that is coming bcoz of shifting of roots concept ,,,,
by graph we can visualise that if the roots are displaced ,the whole function is displaced......
also if we put the roots .the cubic reduues to
ap3+bp2+cp+d which is equal to 0
as p is one of the roots of previos cubic....
62can you expain the last step...??
I dont mean to say that it is wrong :P
1-a(\frac{x+d}{c})^{3}+b((\frac{x+d}{c})^{2})-c(\frac{x+d}{c})+d=0continuing with sankar's solution:
one root is -p2(q+r) say \alpha
now wen we observe this -p.p(q+r)
we get an idea that c/a =(p(q+r)+qr)
and -d/pa=qr
cp=(-\alpha -d) or \alpha =- (pc +d) other roots will also be
-( qc +d)
and -(rc +d)
hence the required cubic is -a(\frac{x+d}{c})^{3}+b((\frac{x+d}{c})^{2})-c(\frac{x+d}{c})+d=0
1@aveek
i wanna see the full soln
if u would be kind enough to post it....
:)
1Sir pura soln. post karna hoga kya ??? Bohut type karna parega
If u say yes to mein apna jaan laga dunga
1Yup got it !!!!!! Guys use my method...the sum's becoming very simple !!! Only put symmetric terms and please no dirty calcs.......the sum and products hav to be individually found out but that's easy with the above posted method .... :-)
1Sir, can i proceed like this :
p3 + b/a p2 = -(cp+d)/a ??
Actually I havn't reg. for AITS....so don hav da soln.
62this is a great first step sankara..
but there is more work to be done.. in terms of expressing the sum of roots, product of roots etc..
and there is a "far" better solution.
3-p2(q+r),-q2(r+p),-r2(q+p)
i guess the rest can be solved if we find sum of roots ,product of roots,and sum of product of two roots, and we know the values of
p+q+r,pq+qr+rp,pqr
if something more work needs to be done pls reply.
62This is the first step for most questions on polynomials :D
and I think a solution can be given this way too...
Aside:(the roots are already given as pqr.. why make them alpha beta and gamma :P)
24Sir are u sure what u wrote in last is OK ???wont the roots be cyclic ??
49if 3 roots are α,β,γ,
αβγ=d/a
αβ+βγ+γα=c/a
α+β+γ=b/a
am i proceeding correctly????
1yup i am very weak at cubics.....open confession..thats why i usually leave them the instant i see the word cubic.....
did a cubic question correctly only once....
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62The chapters that you dont like are the ones you are weak at (atleast that is what happens to me :D)
1it seems like i have seen this one.....anyways fiitjee keeps repeating some questions....
i cant think of any ways to do it just now....
i have never liked cubics much....
62no they are not karna..
For class XI paper.. (A student of mine asked me this yesterday )