Find an nth-degree polynomial function with real coefficients satisfying to the given conditions. If you are using a graphing utility, use it to

Question

Find an nth-degree polynomial function with real coefficients satisfying to the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value

n=3;

4 and 2 i are zeros;

f (2)= -32

f(x)= ?

Answer 1

4, 2i and -2i are zeros (imaginary zeros come in conjugate pairs)

the factors are (x-4)(x^2 + 4) = x^3 -4x^2 +4x -16

f(2)=-32

(-2)^3 -4(-2) + 4(-2) -16 = -8+8 -8-16 = -24

to make -24 into -32 multiply by 4/3. (4/3)(-24) = - 32

f(x) = (4/3)(x^3-4x^2 +4x-16) = 4x^3/3 - 16x^2/3 + 16x/3 - 64/3

if you graph it, the only zero you can verify is 4

4(4^3)/3 -16(4)^2/3 + 16(4)/3 - 64/3 = 256/3 - 256/3 + 64/3-64/3 = 0 or graphically, it's the point (4,0)

the graph only intersects the x axis once. the graph is only of real numbers. The imaginary zeros don't show up in a graph. A graph won't have points (2i,0) or (-2i,0)