What Connections Are There Between Number Theory And Partial Differential Equations?

THEORY OF EQUATIONS :1 ) A)Find the number of real roots to the following equation ? OR?

• B) Factorise the given equation to TWO CUBIC FACTORS or ONE LINEAR FACTOR and "FOURH DEGREE FACTOR". x^6 - 5x^4 + 16x^2 - 72x + 9= 0,,,, Can some given me a solution which not consume a lot of time ? 2nd QUESTION : A)Find all possible roots,real as well as complex,if both exist of the following equation Or B)Find number of real roots to the given equation (97-x)^1/4 + x^1/4 = 5. I have put each of the 2 questions in 2 different ways(A & B).i would want some one to answer A * B individually ,not keeping in mind "A" while answering "B" and vice versa. Source:- MASTER OF MATHEMATICS,ENTRANCE EXAMS ,ISI,KOLKTA(INDIA) Thanking everyone for the effort in advance.. Please do help.

• Answer:

  1/ A The second derivative is 6*5 x^4 - 5*4*3 x^2 + 16*2. Since the discriminant is < 0, the graph is convex and there are at most 2 roots. P(0) >0 and P(1)<0. So you have 2 roots. So if I get it well, if you answer A you can forget about B? 2/ B The function is concave. So there are only 2 roots: 16 and 81 giving 2+3 = 3+2 = 5.