Construct a polynomial function with the following properties: fifth degree, −2 is a zero of multiplicity 2, 2 is the only other zero.

Answer 1

There are LOTS of possible solutions to this problem. If x = -2 (multiplicity of 2) is a zero then (x + 2) is a factor twice. If x = 2 is the only other real zero then (x - 2) is a factor. Since this is a degree 5 polynomial, that means (since there are 3 real zeros) there must be 2 complex zeros. Let's say one of them is "i", then by the conjugate zero theorem, "-i" is also a zero. (note that you can pick ANY complex zeros as long as they are conjugates)

So the zeros are -2, -2, 2, i, -i meaning the factors are (x + 2)(x + 2)(x - 2)(x - i)(x + i) are the factors of the polynomial. To get the "multiplied out" polynomial, just multiply these all together.