division of 2 equations

Is the result of the division of 2 equations is another equation?

Suppose we hove (ax^b+cx^d)/(ex^f+gx^h)

So is the result wel be another equation with different factor and power like (mx^n+sx^p) or will be something els.

Answer 1

[Walter Roberson]

It will be "something else".

The question is whether you can rationalize the division of two polynomials to create a polynomial. The answer to that is No, not generally.

Consider for example,

   fplot(@x) x ./ (x + 1), [-5 5])

This has a discontinuity at x = -1, being positive to the left and negative to the right, with asymptotic 0 at either side. This is clearly not a polynomial.

Answer 2

[marwan mokbil]

Thanks for your answer. But what do you mean by some thing els. My question if you have equation like (3x+2)/(5x-6) is the result can be some thing like ax^b+cx^d+.... I mean the result will not be again rational. Is that possible or sill the result we be rational.

[Walter Roberson]

fplot(@(x) (3*x+2)./(5*x-6), [-10 10])

This is not a polynomial in x.

Is it possible to find a polynomial that is in some sense similar? Sort of, at least in this case:

syms x z
f = 3/5 + 28/25*z
subs(f, z, 1/(x-6/5))

gives you your (3*x+2)/(5*x-6), just in a different form, so f is a related polynomial, through a change of variables involving a rational but non-linear function of x. But this is not as simple as x = a*y^n for some positive integer n -- that is, you cannot just write (3*x+2)/(5*x-6) in terms of a0*x^n + a1*x^(n-1) + ... an*x + an1 + an2*x^(-1) + an3*x^(-3) ... for some finite sequence of coefficients.

It is possible to convert to an infinite series

>> series((3*x+2)/(5*x-6),x,'Order',10)
ans =
- (7*x)/9 - (35*x^2)/54 - (175*x^3)/324 - (875*x^4)/1944 - (4375*x^5)/11664 - (21875*x^6)/69984 - (109375*x^7)/419904 - (546875*x^8)/2519424 - (2734375*x^9)/15116544 - 1/3

Although this is polynomial in form, this is a truncation of an infinite series that is wrong infinitely often.