A polynomial $$ f(x) $$ and two of its zeros are given. $$ f(x)=4 x^{5}+23 x^{4}+46 x^{3}+3 x^{2}-100 x+24 $$; $$ -2-2 i $$ and $$ \frac{1}{4} $$ are zeros Part: $$ 0 / 3 $$ Part 1 of 3 (a) Find all the zeros. Write the answer in exact form. If there is more than one answer, separate them with commas. Select "None" if applicable. Ds The zeros of $$ f(x) $$ :

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The given polynomial is
$$
f(x)=4x^{5}+23x^{4}+46x^{3}+3x^{2}-100x+24.
$$
We are given two zeros: 
$$
-2-2i \quad \text{and} \quad \frac{1}{4}.
$$
Since the coefficients of $$ f(x) $$ are real, the complex conjugate of $$-2-2i$$, namely
$$
-2+2i,
$$
must also be a zero.

First, note that the zero $$\frac{1}{4}$$ corresponds to the factor $$4x-1$$, so we can write
$$
f(x) = (4x-1)Q(x),
$$
where $$ Q(x) $$ is a degree 4 polynomial.

Performing polynomial long division of $$ f(x) $$ by $$ 4x-1 $$ yields:
$$
Q(x)=x^4+6x^3+13x^2+4x-24.
$$

Now, we know that $$-2 \pm 2i$$ are zeros. The quadratic factor with these zeros is found by writing it in the form:
$$
x^2+4x+8,
$$
since the sum of the roots is $$-4$$ and their product is $$8$$.

Divide $$ Q(x) $$ by $$ x^2+4x+8 $$. Assume
$$
Q(x) = (x^2+4x+8)(ax^2+bx+c).
$$
Since the coefficient of $$ x^4 $$ in $$ Q(x) $$ is 1, we have $$ a=1 $$. Expanding, we get:
$$
(x^2+4x+8)(x^2+bx+c) = x^4 + (b+4)x^3 + (c+4b+8)x^2 + (4c+8b)x + 8c.
$$
Matching coefficients with
$$
x^4+6x^3+13x^2+4x-24,
$$
we obtain:
1. $$ b+4 = 6 $$ $$\Longrightarrow$$ $$ b=2 $$.
2. $$ c+4b+8 = c+8+8 = c+16 = 13 $$ $$\Longrightarrow$$ $$ c=-3 $$.
3. Check: $$ 4c+8b = 4(-3)+8(2) = -12+16=4 $$ and $$ 8c=-24 $$.

Thus,
$$
Q(x) = (x^2+4x+8)(x^2+2x-3).
$$

Now, factor the quadratic $$ x^2+2x-3 $$ further:
$$
x^2+2x-3 = (x+3)(x-1).
$$

Thus, the complete factorization of $$ f(x) $$ is:
$$
f(x)= (4x-1)(x^2+4x+8)(x+3)(x-1).
$$

The zeros of $$ f(x) $$ are then:
$$
x=\frac{1}{4},\quad -2+2i,\quad -2-2i,\quad -3,\quad 1.
$$
The zeros of $$ f(x) $$ are $$ \frac{1}{4}, -2+2i, -2-2i, -3, 1 $$.

A polynomial and two of its zeros are given.
;
and are zeros
Part:
Part 1 of 3
(a) Find all the zeros. Write the answer in exact form.
If there is more than one answer, separate them with commas.
Select “None” if applicable. Ds
The zeros of :

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A polynomial and two of its zeros are given. ; and are zeros Part: Part 1 of 3 (a) Find all the zeros. Write the answer in exact form. If there is more than one answer, separate them with commas. Select “None” if applicable. Ds The zeros of :