Final answer:
The polynomial function with degree 4, a y-intercept of 12, and roots at -3, 4, and 1 (with multiplicity 2) is f(x) = -(x + 3)(x - 4)(x - 1)^2.
Step-by-step explanation:
The polynomial function in question, f(x), has a degree of 4, a y-intercept of 12, roots at -3 and 4, and a root of 1 with multiplicity 2. To construct this polynomial function, we use the fact that the roots of a polynomial are the values of x for which the polynomial equals zero. The root of 1 with multiplicity 2 signifies that the factor associated with this root will be squared in the polynomial function.
Starting with the roots, the factors of the polynomial will be:
- (x + 3) for the root at -3
- (x - 4) for the root at 4
- (x - 1)^2 for the root of 1 with multiplicity 2
Thus, the polynomial function is:
f(x) = a(x + 3)(x - 4)(x - 1)^2
Now, to find the coefficient a, we use the y-intercept of 12. A y-intercept of 12 means that f(0) = 12. Therefore:
12 = a(0 + 3)(0 - 4)(0 - 1)^2
12 = a(3)(-4)(1)
12 = -12a
a = -1
Substituting a back into the polynomial function, we get:
f(x) = -1(x + 3)(x - 4)(x - 1)^2
f(x) = -(x + 3)(x - 4)(x - 1)^2