Write a polynomial \( f(x) \) that meets the given conditions. Answers may vary. Degree 6 polynomial with zeros of \( 3, \frac{1}{2} \) (each with multiplicity 1 ), and 0 (with multiplicity 4). \( f(x)= \)

To construct the polynomial \( f(x) \), we can use the given zeros. A polynomial is formed by multiplying factors corresponding to its zeros. Since the zero \( 0 \) has a multiplicity of 4, the factor for this root will be \( x^4 \). For the zero \( 3 \) (with multiplicity 1), the factor is \( (x - 3) \), and for the zero \( \frac{1}{2} \) (also with multiplicity 1), the factor is \( \left(x - \frac{1}{2}\right) \). Putting it all together, we have: \[ f(x) = x^4 (x - 3) \left(x - \frac{1}{2}\right) \] Now, let's expand this. First, multiply the linear factors: \[ (x - 3) \left(x - \frac{1}{2}\right) = x^2 - \frac{1}{2}x - 3x + \frac{3}{2} = x^2 - \frac{7}{2}x + \frac{3}{2} \] Next, multiply this quadratic by \( x^4 \): \[ f(x) = x^4 \left(x^2 - \frac{7}{2}x + \frac{3}{2}\right) = x^6 - \frac{7}{2}x^5 + \frac{3}{2}x^4 \] Thus, the polynomial \( f(x) \) can be succinctly expressed as: \[ f(x) = x^6 - \frac{7}{2}x^5 + \frac{3}{2}x^4 \] Feel free to write it in a different form or check the expansion for any inconsistencies!