Final answer:
To find the equation of a cubic polynomial with zeros -5, 4, 5, and a y-intercept of -200, start by writing the factors associated with the zeros, determine the leading coefficient using the y-intercept, and then simplify the expression to get the final polynomial f(x) = 2x^3 - 12x^2 + 10x + 200.
Step-by-step explanation:
To write the equation of a cubic polynomial with zeros -5, 4, and 5 and a y-intercept of -200, you would start by using the fact that the zeros of the polynomial give us factors of the polynomial. The factors corresponding to the zeros -5, 4, and 5 are (x + 5), (x - 4), and (x - 5) respectively. Therefore, the polynomial can be written as f(x) = a(x + 5)(x - 4)(x - 5).
To find the value of 'a', we use the y-intercept. The y-intercept occurs when x = 0, so we substitute x = 0 into f(x) and set it equal to -200: -200 = a(0 + 5)(0 - 4)(0 - 5). Solving for 'a' gives us a = -200/(-100) = 2. Now we have the value of 'a', so we can write the full polynomial equation: f(x) = 2(x + 5)(x - 4)(x - 5).
Finally, we simplify the polynomial by expanding the terms: f(x) = 2[(x2 - 5x + 4x - 20)(x - 5)] = 2[x2 - x - 20)(x - 5)] = 2[x3 - 5x2 - x2 + 5x - 20x + 100]
f(x) = 2x3 - 12x2 + 10x + 200.