Answer:
f(x) = 2(x - 2)²(x + 1)(x - 1)
Explanation:
a polynomial of the 4th degree has as highest exponent for terms in x : 4
your previous attempt only gets you to x³ terms, so to a polynomial of 3rd degree.
you made the following mistakes :
you used the factor for the root (zero) at x = 2 only once, although it has a multiplicity 2 (that means it has to be used twice : the same zero solution appears twice) :
e.g. (x - 2)²
you used the same factor for the root at -1 and at +1.
no, they are different, of course, as -1 and +1 are different.
e.g.
(x + 1)(x - 1)
you miscalculated the factor to fulfill also the last criteria to have a y- intercept at (0, -8), y = f(0) = -8.
so, the equation is
f(x) = 2(x - 2)²(x + 1)(x - 1)
why ?
when x = 0, all terms are eliminated except for the last one that is the multiplication of all constant integers of the factors. and to get -8 in such a multiplication, we have already 3 negative factors (that will create a negative result), and any other factor needs to be positive to keep that negative result.
as -2, -2 (due to the ² factor we have it twice in the calculation), +1 and -1 are necessary for the roots (giving us -4 as result), we need an initial integer factor to now create the needed -8 result (2 × -4 = -8) :
2 × -2 × -2 × +1 × -1