Write an expanded polynomial equation based on the following information.
A cubic function with a zero at x=3 (with multiplicity 2), a zero at x=2, and a constance term of 18.
f(x) = (x-3)^2(x-2)+c
Note that 3^2*2 = 18
But that will be -18 (why?), so c=36
and that's "constant" not constance.
To write the expanded polynomial equation, we need to know the factors of the equation. Given the information, we can deduce the following factors:
1. A zero at x=3 with multiplicity 2 means that (x-3) is a factor twice: (x-3)(x-3) = (x-3)^2.
2. A zero at x=2 means that (x-2) is another factor.
To find the constant term, we set x=0 in the polynomial equation. Since the constant term is given as 18, we have:
(x-3)(x-3)(x-2) = 18.
Now, to expand the equation, we multiply the factors together:
(x-3)(x-3)(x-2) = 18
(x^2 - 6x + 9)(x-2) = 18
(x^2 - 6x + 9)x - (x^2 - 6x + 9)(2) = 18
x^3 - 6x^2 + 9x - 2x^2 + 12x - 18 = 18
x^3 - 8x^2 + 21x - 18 = 18
In the last step, we combined like terms. The expanded polynomial equation is:
x^3 - 8x^2 + 21x - 36 = 0.