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find a degree 3 polynomial with real coefficients having zeros 2 and 4i and a lead coefficient of 1. write p in expanded form.
p(x) =
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Explanation:

Step1: Identify all zeros

For real coefficients, complex zeros come in conjugate pairs. If $4i$ is a zero, then $-4i$ is also a zero. The zeros are $2$, $4i$, $-4i$.

Step2: Write polynomial in factored form

With leading coefficient 1, the factored form is:
$P(x)=(x-2)(x-4i)(x+4i)$

Step3: Multiply complex factors first

Use difference of squares: $(x-4i)(x+4i)=x^2-(4i)^2$
$x^2-(4i)^2 = x^2 - 16i^2 = x^2 + 16$ (since $i^2=-1$)

Step4: Multiply by remaining factor

$P(x)=(x-2)(x^2+16)$
Expand using distributive property:
$P(x)=x(x^2+16)-2(x^2+16)=x^3+16x-2x^2-32$

Step5: Rearrange into standard form

$P(x)=x^3-2x^2+16x-32$

Answer:

$x^3-2x^2+16x-32$