The polynomial of degree \( 3, P(x) \), has a root of multiplicity 2 at \( x=4 \) and a root of multiplicity 1 at \( x=-4 \). The \( y \)-intercept is \( y=-19.2 \). Find a formula for \( P(x) \). \( P(x)= \)

To find the polynomial \( P(x) \), we start by using the information about the roots. Since \( x = 4 \) is a root of multiplicity 2, we can express \( P(x) \) as \( P(x) = k(x - 4)^2(x + 4) \) where \( k \) is a constant that we need to determine. Next, we use the given \( y \)-intercept, which occurs when \( x = 0 \): \[ P(0) = k(0 - 4)^2(0 + 4) = k(16)(4) = 64k \] We know that this equals -19.2, so we can set up the equation: \[ 64k = -19.2 \] Now, we can solve for \( k \): \[ k = \frac{-19.2}{64} = -0.3 \] Now substituting \( k \) back into the polynomial formula, we have: \[ P(x) = -0.3(x - 4)^2(x + 4) \] Expanding this, we can first expand \( (x - 4)^2 \): \[ (x - 4)^2 = x^2 - 8x + 16 \] Now multiply this by \( (x + 4) \): \[ (x^2 - 8x + 16)(x + 4) = x^3 + 4x^2 - 8x^2 - 32x + 16x + 64 = x^3 - 4x^2 - 16x + 64 \] Finally, applying the constant \( -0.3 \): \[ P(x) = -0.3(x^3 - 4x^2 - 16x + 64) = -0.3x^3 + 1.2x^2 + 4.8x - 19.2 \] Therefore, the formula for \( P(x) \) is: \[ \boxed{-0.3x^3 + 1.2x^2 + 4.8x - 19.2} \]