Find a polynomial of degree 3 with real coefficients and zeros of -3, -1, and 4, for which f(-2)=24

We want to find a polynomialf(x) = a x³ + b x² + c x + dsuch that the roots of f are x = -3, x = -1, and x = 4, and f(x) takes on a value of -24 when x = -2.The factor theorem for polynomials tells us that we can factorize f(x) asa x³ + b x² + c x + d = a (x + 3) (x + 1) (x - 4)Expand the right side:(x + 3) (x + 1) (x - 4) = x³ - 13x - 12So we havea x³ + b x² + c x + d = a x³ - 13a x - 12aIn order for both sides to be equal, the coefficients of both polynomials on terms of the same degree must be equal. This meansa = a (of course)b = 0 (there is no x² term on the right)c = -13ad = -12aWe also have that f (-2) = -24, which meansf (-2) = a (-2 + 3) (-2 + 1) (-2 - 4)-24 = 6aa = -4which in turn tells us that c = 52 and d = 48.So we foundf(x) = -4x³ + 52x + 48