Create a polynomial \( p \) that has the desired characteristics. You may leave the polynomial in factored form. The zeros of \( p \) are -4 and 2 . - \( x=-4 \) is a zero of multiplicity 2 . - \( p(x) \) has \( y \)-intercept \( (0,-192) \) is degree 3 . \( p(x)= \)

To create the polynomial \( p \) with the given characteristics, let's start with the zeros. Since \( x = -4 \) is a zero of multiplicity 2, this means we will include \( (x + 4)^2 \) in our polynomial. Also, since there is another zero at \( x = 2 \), we will include \( (x - 2)^1 \). Putting it all together, the polynomial in factored form is: \[ p(x) = k (x + 4)^2 (x - 2) \] where \( k \) is a constant that we will determine based on the \( y \)-intercept. To find \( k \), we know that the \( y \)-intercept is \( (0, -192) \). This means we need to calculate \( p(0) \): \[ p(0) = k(0 + 4)^2(0 - 2) = k(4^2)(-2) = k \cdot 16 \cdot (-2) = -32k \] We want \( p(0) = -192 \), so we can set up the equation: \[ -32k = -192 \] Solving for \( k \), we divide both sides by -32: \[ k = \frac{-192}{-32} = 6 \] Now we can substitute \( k \) back into the polynomial: \[ p(x) = 6(x + 4)^2 (x - 2) \] So, the polynomial in its final factored form is: \[ p(x) = 6(x + 4)^2 (x - 2) \]