5. Find the exact number of real zeros and imaginary of the polynomial $x^9 + 9x^7 + 7x^5 + 5x^3 + 3x$
Soln:
Let $P(x) = x^9 + 9x^7 + 7x^5 + 5x^3 + 3x$
No. of sign changes $= 0$
No positive zero$P(-x) = -x^9 - 9x^7 - 7x^5 - 5x^3 - 3x$
No. of sign changes $= 0$
No negative zero.$P(0) = 0$
$0$ is a zero.Among 9 zeros, no positive zero, no negative zero but one zero
Hence given expression has 8 imaginary.