Questions: A polynomial P is given. P(x)=x^4+64 x^2 (a) Find all zeros of P, real and complex. x=

Solution Steps

To find the zeros of the polynomial $P(x) = x^4 + 64x^2$, we can factor the polynomial and solve for $x$. First, factor out the common term $x^2$, then solve the resulting quadratic equation.

The polynomial $P(x) = x^4 + 64x^2$ can be factored as follows: $$P(x) = x^2(x^2 + 64)$$

To find the zeros of $P(x)$, we set the factored form equal to zero: $$x^2(x^2 + 64) = 0$$ This gives us two equations to solve:

1. $x^2 = 0$
2. $x^2 + 64 = 0$

From the first equation $x^2 = 0$, we find: $$x = 0$$

From the second equation $x^2 + 64 = 0$, we solve for $x$: $$x^2 = -64 \implies x = \pm 8i$$

Final Answer