Find a polynomial function of degree 4 with -3 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1 . The polynomial function in expanded form is \( f(x)=\square \) (Use 1 for the leading coefficient.)

To construct the polynomial function, start with the zeros given: -3 (with a multiplicity of 3) and 0 (with a multiplicity of 1). The polynomial can be expressed in factored form as: \[ f(x) = (x + 3)^3 \cdot x \] Now, let's expand this expression step by step. First, expand \( (x + 3)^3 \): \[ (x + 3)^3 = (x + 3)(x + 3)(x + 3) \] Calculating \( (x + 3)(x + 3) \) first: \[ (x + 3)(x + 3) = x^2 + 6x + 9 \] Now we multiply this result by \( (x + 3) \): \[ (x^2 + 6x + 9)(x + 3) = x^3 + 3x^2 + 6x^2 + 18x + 9x + 27 = x^3 + 9x^2 + 27x + 27 \] Next, we multiply this expansion by \( x \) to include the zero at 0: \[ f(x) = x(x^3 + 9x^2 + 27x + 27) = x^4 + 9x^3 + 27x^2 + 27x \] Thus, the polynomial function in expanded form is: \[ f(x) = x^4 + 9x^3 + 27x^2 + 27x \]