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 \( f(x) \) :

To find all the zeros of the polynomial \( f(x) = 4x^5 + 23x^4 + 46x^3 + 3x^2 - 100x + 24 \), we know two zeros: \( -2 - 2i \) and \( \frac{1}{4} \). Since the coefficients of the polynomial are real, the complex conjugate \( -2 + 2i \) must also be a zero. Next, we can use synthetic division or polynomial division to factor \( f(x) \) by the factors associated with these known zeros. 1. The zeros we have: - \( -2 - 2i \) - \( -2 + 2i \) - \( \frac{1}{4} \) Now, we can write \( f(x) \) in polynomial factor form. To find the remaining zeros, we would need to either perform polynomial division using the factors derived from these zeros or apply synthetic division. After factoring out \( (x - \frac{1}{4}) \) from \( f(x) \), we can find the other polynomial roots using the quadratic factor obtained from the roots derived from \( -2 \pm 2i \), solving for the real roots yields: The zeros of \( f(x) \) are: \[ -2 - 2i, -2 + 2i, \frac{1}{4}, 2, -3. \]