no of postive integral solns

Thanks Debotosh :)

which one do you want to rename karna?

We expect a cubic equation to have 3 solutions since the highest power of x is 3. In the following image, we see that the curve cuts the x-axis in 3 places, giving us 3 roots.

But sometimes, there is a repeated root, so we only have 2 places where the curve cuts the x-axis.

An example here could be (x − 1)(x − 3)2 = 0.

Back to our first cubic equation example, x3 + x2 − x + 2 = 0. The mathematicians could find one solution OK (x = −2), but where were the other 2?

The graph of this situation is as follows. There is one point where the curve cuts through the x-axis.

I din have the tempo to type the whole thing so searched the net and copy pasted some stuff to show the behavior of the cubic equation .. I hope this explains some of the not so obvious things in the above post.

@debotosh .. yes

@madhu..

see the way to approach this question is that x^3+2x^2+1 has its derivative as 3x^2+4x....

which means the derivative is zero at the points x=0 and x=-4/3

let us see the value of the function at x=0.. it is 0^3+2.0^2+1=0 and at x=-4/3 the value of the function is -64/27+31/9+1 which is +ve .

hence the value of the function is +ve at both these roots..
so the number of roots is 1

Here keep in mind the shape of the cubic...
it rises, comes down and rises again (except in a lot of cases like x^3 where it just keeps rising monotonically)

xyz .. after doing this,

the fact is that every root will be a factor of 440.....

the number of factors is not very large and hence solving the cubic should not be a big issue

also what you can interpret is that if you want all x,y and z to be integers, that two of x, y and z are of the form 2n+1 while one is of form 2n

no madhu

try to look only at the first derivative

and read my post agian..

One more thing that i forgot to mention is that you can look at the second derivative to find the points where the graph has changed.. (I mean the derivative becomes zero)

The points can be found out because the derivative will give a quadratic...

If that does not have a real root then the cubic has only one real root.

if it has real roots which are distinct then the cubic may have one or 3 real roots depending on what the values of the function is at these two points...

For instance tell me the number of real roots of

x^3+2x^2+1= 0 (Try not to use descarte's rule of sign change)

Then for cubics there is a direct formula.. which i will nto be able to learn in my life.. nor is it advisable

but for a lot of such problems at our level, what we should do is try and see if 1, -1 or zero becomes one of the roots

sometimes in some trick questions, substituing x = cos t gives us cos (3t)

so even that is something that can be kept in mind... there was one JEE question sometime on this too.....

yes if it known to have integer solutions then it would not be a bad starting point to guess...