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

Transcript text: A polynomial $P$ is given. \[ P(x)=x^{4}+64 x^{2} \] (a) Find all zeros of $P$, real and complex. \[ x=\square \]

Solution

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.

Step 1: Factor the Polynomial

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

Step 2: Solve for Zeros

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:

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

Step 3: Find Real and Complex Solutions

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

The zeros of the polynomial $ P(x) $ are: \[ \boxed{0, -8i, 8i} \]

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