Polynomials

Question

Find a polynomial with integer coefficients, with leading coefficient 1, degree 5, zeros i and 6- i, and passing through the origin?

Answer 1

P(x) = x^5 + Ax^4 + Bx^3 + Cx^2 + Dx + E

 

  Passing through the origin forces E=0

 

P(x) = x^5 + Ax^4 + Bx^3 + Cx^2 + Dx

 

Since i and 6-i and zeros, then so are -i and 6+i

 

x(x-i)(x+i)(x - (6+i)) ( x - (6-i))=0

 

x(x^2+1)(x^2 - x(6-i) - (6+i)x + (36+1)) = 0

x(x^2+1)(x^2 -12x + 37) = 0