Write a polynomial \( f(x) \) that satisfies the given conditions. Polynomial of lowest degree with zeros of -4 (multiplicity 1 ), 1 (multiplicity 1 ), and with \( f(0)=16 \). \( f(x)=\square \)

To create a polynomial \( f(x) \) with the specified zeros, we start by incorporating the zeros into the polynomial's basic form. Given the zeros of -4 and 1, we can express the polynomial as: \[ f(x) = k(x + 4)(x - 1) \] Here, \( k \) is a constant that we need to determine in order to satisfy \( f(0) = 16 \). Now, we can plug in \( x = 0 \): \[ f(0) = k(0 + 4)(0 - 1) = k(4)(-1) = -4k \] Setting this equal to 16 gives us: \[ -4k = 16 \] Dividing both sides by -4 yields: \[ k = -4 \] Now we can write the polynomial: \[ f(x) = -4(x + 4)(x - 1) \] Expanding this, we first calculate \( (x + 4)(x - 1) = x^2 + 3x - 4 \). Now multiplying by -4 results in: \[ f(x) = -4(x^2 + 3x - 4) = -4x^2 - 12x + 16 \] Therefore, the polynomial you are looking for is: \[ f(x) = -4x^2 - 12x + 16 \]