How can I solve this problem?

Question

Find a 5th degree polynomial P, with integer coefficients, that has 4 as its leading coefficient, and whose list of zeros include 1/2, -i, and -1 is a zero of multiplicity 2.

Answer 1

4(x-1/2)(x^2+1)(x+1)^2, multiplied out gives a 5th degree polynomial with leading coefficient of 4x^5

4x^5 +6x^4+4x^3 +12x^2 +8x -2

x=1/2 becomes (x-1/2)=0, which makes x-1/2 a factor of the polynomial

x=i becomes x-i=0 x-i is a factor, so is x+i as they come in conjugate pairs for imaginary zeros. (x-i)(x+i) = x^2+1

x=-1 becomes x+1=0, so x+1 is another factor.

Put them all together as a product. To get a leading coefficient of 4, just multiply by 4