If (2x-1)^20 - (ax+b)^20 = (x^2 + px +q)^10, then find the value of a, b, p and q.

Question

This question is asked in polynomials

Answer 1

If a, b, p, q are all rational then there can be no solution. Here's why:

Because this is an equation, the left and right sides must balance. If we take the coefficients of the highest degree of x (x²⁰) we have the equation 2²⁰-a²⁰=1, so a²⁰=2²⁰-1.  a would therefore be irrational. a would also be very close to 2 (about 1.999999905).

Similarly if we take the constant term on both sides we get the equation 1-b²⁰=q¹⁰, 0≤b²⁰≤1, since q¹⁰ has to be positive (raised to an even power). If b=1, q=0 and if b=0, q=1. Other fractional values of b would make q an irrational fraction. For example, if b=0.9, q=0.98712 approx.