Find the number of real roots of \(2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \dots + 2x + 3 = 0.\)
+128475 2x^2001 + 3x^2000 + 2x^1999 + 3x^1998 + .....+ 2x + 3 note that we can write
x^2000 ( 2x + 3) + x^1998(2x + 3) + x^1996(2x + 3) + ....+ x^2(2x + 3) + x^0 (2x + 3) =
x^2000(2x+ 3) +x^!998(2x + 3) + x^1996(2x + 3) + ......+ x^2(2x + 3) + 1(2x + 3) =
(2x + 3) ( x^2000 + x^1998 + x^1996 + .....+ x^2 + 1)
Since the second polynomial is > 0 for all x the only real root is
2x + 3 =0
2x = - 3
x = -3/2
