Final answer:
To find the equation of a fourth degree polynomial function in standard form that goes through the given points (-2,0), (2,0), (-1,0), (3,0), and (0,12), we use the fact that a polynomial of degree n will have n + 1 distinct points. The equation will have the factors (x + 2)(x - 2)(x + 1)(x - 3), resulting in f(x) = x^4 - 5x^2 + 4.
Step-by-step explanation:
To find the equation of a fourth degree polynomial function in standard form that goes through the given points (-2,0), (2,0), (-1,0), (3,0), and (0,12), we can use the fact that a polynomial of degree n will have n + 1 distinct points. Since the given points are all on the x-axis, the equation must have at least four factors of (x - r), where r is the x-coordinate of each point. Therefore, the equation will have the factors (x + 2)(x - 2)(x + 1)(x - 3).
Multiplying these factors gives us the equation (x^2 - 4)(x^2 - 1) = x^4 - 5x^2 + 4.
So, the equation of the fourth degree polynomial function in standard form that goes through the given points is f(x) = x^4 - 5x^2 + 4.