Polynomial function (linear factorization + descarter's rule)

Question

Find an ntn-degree polynomial function with real coefficients satisfying the given conditons. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=4; -1, 3, and 2 + 3 i are zeros; f(1)= -120

Answer 1

The zeros given can be used to come up with the factors. It would look like the following:

f(x)=a(x-4)(x+1)(x-3)(x-2+3i)(x-2-3i) FOIL the last 2 factors

f(x)=a(x-4)(x+1)(x-3)(x²-4x+13)

Using the last bit of information, we can solve for a coefficient, a.

-120=a(1-4)(1+1)(1-3)(1²-4(1)+13) simplify

-120=a(-3)(2)(-2)(10)

-120=120a divide both sides by 120

a = -1

Plug in this value for a,

f(x) = -(x-4)(x+1)(x-3)(x²-4x+13) You should multiply these all together to get the polynomial in its final form.