Say I have 2 quadratics
p(x^2)+qx=r=0 and d(x^2)+ex+f=0 with one common root
can i write (p/d)=(q/e)=(r/f) ?
If $\displaystyle p(x),q(x)$ are two polynomials with the exact same set of roots and the same degree, then $\displaystyle p(x) = kq(x)$ where $\displaystyle k$ is some constant. That is when you can compare coefficients.
hey
like comparing coefficients.
can we compare exponents as in this given equation
[(a^n) + (b^n)]/[(a^n-1) + (b^n-1)] = (a+b)/2 (n-1 is exponent)
to get n=1
if yes then what is the condition that allows us to compare?
Maybe this will help
Two quadratic polynomials
$\displaystyle x^2 + b x + c$
$\displaystyle x^2 + s x + t$
have a root in common if and only if
$\displaystyle c^2 - b c s + c s^2 + b^2 t - 2 c t - b s t + t^2 = 0$
Well...We can solve this in terms of a and b: b = -a for odd n, so there is no specific relationship needed for n. But generally we can match coefficients of terms on each side of an equation if the terms are members of an orthogonal series. The "power series" is such a series: $\displaystyle a_0 + a_1x + a_2 x^2 + ...$. Many other orthogonal expansions exist but you'll see the power series the most often.
-Dan