• Oct 31st 2013, 08:17 PM
AaPa

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) ?
• Oct 31st 2013, 08:31 PM
topsquark
Quote:

Originally Posted by AaPa

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) ?

Can you?

Compare: $\displaystyle 2x^2 + x - 1$ vs. $\displaystyle 2x^2 + 3x - 2$

-Dan
• Oct 31st 2013, 10:43 PM
AaPa
no.
so when can we compare coefficients?
• Oct 31st 2013, 10:46 PM
SlipEternal
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.
• Nov 1st 2013, 02:18 AM
AaPa
All my confusions which troubled me have disappeared. You really made me think on my own.

Thank you very very muchh!!
• Nov 1st 2013, 02:24 AM
AaPa
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?
• Nov 1st 2013, 10:03 PM
AaPa
$a^n + b^n / a^n-1 +b^n-1 = a+b / 2$
• Nov 2nd 2013, 12:06 AM
Idea
Maybe this will help

$\displaystyle x^2 + b x + c$

$\displaystyle x^2 + s x + t$

have a root in common if and only if

$\displaystyle c^2 - b c s + c s^2 + b^2 t - 2 c t - b s t + t^2 = 0$
• Nov 2nd 2013, 04:26 AM
AaPa
$\displaystyle \frac {a^n + b^n} { a^{n-1} + b^{n-1}} = \frac{a+b} {2}$

by comparing exponents we get n=1
exactly when can we compare exponents like this?
• Nov 2nd 2013, 05:42 AM
topsquark
Quote:

Originally Posted by AaPa
$\displaystyle \frac {a^n + b^n} { a^{n-1} + b^{n-1}} = \frac{a+b} {2}$

by comparing exponents we get n=1
exactly when can we compare exponents like this?

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
• Nov 2nd 2013, 08:42 AM
AaPa