The zeros of a quartic function are -3, -1, and 2 (order 2). Determine
a). an equation for the function that satisfies this condition.
b). an equation for a function that satisfies this condition and passes through the point (1, 4)
a). Since the zeros are -3, -1, and 2 (order 2), the factors of the quartic function are (x + 3), (x + 1), and (x - 2)^2. Multiplying these factors together we get:
f(x) = (x + 3)(x + 1)(x - 2)^2
f(x) = (x^2 + 4x + 3)(x^2 - 4x + 4)
f(x) = x^4 + 4x^3 + 3x^2 - 4x^3 - 16x^2 - 12x + 4x^2 + 16x + 12
f(x) = x^4 - 9x^2 + 12
Therefore, an equation for the function that satisfies the given conditions is f(x) = x^4 - 9x^2 + 12.
b). To find an equation for a function that passes through the point (1, 4), we can add the point (1,4) to the equation and solve for the constant term:
4 = (1)^4 - 9(1)^2 + 12 + C
4 = 1 - 9 + 12 + C
4 = 4 + C
C = 0
Therefore, the function that satisfies the given conditions and passes through the point (1, 4) is f(x) = x^4 - 9x^2 + 12.