Given $$ P(x)=3 x^{5}+2 x^{4}+41 x^{3}+34 x^{2}-112 x+32 $$, and that $$ 4 i $$ is a zero, write $$ P $$ in factored form (as a product of linear factors). Be sure to write the full equation, including $$ P(x)= $$.

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We start with
$$
P(x)=3x^5+2x^4+41x^3+34x^2-112x+32.
$$
Since $$4i$$ is a zero and the coefficients of $$P$$ are real, its complex conjugate $$-4i$$ must also be a zero. Thus, the factors
$$
(x-4i) \quad \text{and} \quad (x+4i)
$$
appear in the factorization. Their product is
$$
(x-4i)(x+4i)=x^2+16.
$$

Next, we perform the division
$$
\frac{P(x)}{x^2+16}
$$
to find the remaining factor. Dividing step by step:

1. Divide the first term $$3x^5$$ by $$x^2$$ to get $$3x^3$$. Multiply and subtract:
   $$
   3x^5+2x^4+41x^3+34x^2-112x+32 - \left(3x^5+48x^3\right) = 2x^4 -7x^3+34x^2-112x+32.
   $$

2. Divide $$2x^4$$ by $$x^2$$ to get $$2x^2$$. Multiply and subtract:
   $$
   2x^4 -7x^3+34x^2-112x+32 - \left(2x^4+32x^2\right)= -7x^3+2x^2-112x+32.
   $$

3. Divide $$-7x^3$$ by $$x^2$$ to get $$-7x$$. Multiply and subtract:
   $$
   -7x^3+2x^2-112x+32 - \left(-7x^3-112x\right)= 2x^2+32.
   $$

4. Finally, divide $$2x^2$$ by $$x^2$$ to get $$2$$. Multiply and subtract:
   $$
   2x^2+32- \left(2x^2+32\right)=0.
   $$

The quotient is
$$
3x^3+2x^2-7x+2.
$$
Thus, we have
$$
P(x)=(x^2+16)(3x^3+2x^2-7x+2).
$$

Now, we factor the cubic $$3x^3+2x^2-7x+2$$. Testing possible roots, we find:
$$
3(1)^3+2(1)^2-7(1)+2=3+2-7+2=0,
$$
so $$x=1$$ is a zero. Dividing by $$x-1$$ (using synthetic division or polynomial division) gives:
$$
3x^3+2x^2-7x+2=(x-1)(3x^2+5x-2).
$$

Next, we factor the quadratic $$3x^2+5x-2$$. We look for two numbers whose product is $$3(-2)=-6$$ and whose sum is $$5$$. We can write:
$$
3x^2+5x-2=3x^2+6x-x-2=3x(x+2)-1(x+2)=(x+2)(3x-1).
$$

Thus, the complete factorization over the real numbers is
$$
P(x)=(x^2+16)(x-1)(x+2)(3x-1).
$$

Finally, since the problem asks for a product of linear factors (over the complex numbers), we factor $$x^2+16$$ as
$$
x^2+16=(x-4i)(x+4i).
$$

Therefore, the fully factored form is
$$
P(x)=(x-4i)(x+4i)(x-1)(x+2)(3x-1).
$$
$$ P(x) = (x - 4i)(x + 4i)(x - 1)(x + 2)(3x - 1) $$

Given , and that is a zero, write in factored form
(as a product of linear factors). Be sure to write the full equation, including .

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