Find a polynomial function with real coefficients of degree

Find a polynomial function with real coefficients of degree 4 with zeros 1 + i and 2 - 3i. Factor x^4 + x over complex numbers.

Solution

1)given coefficients are real and 1+i , 2-3i are zeroes

as coefficients are real  1-i , 2+3i are also zeroes

polynomial is of form

f(x)=k(x-(1+i))(x-(2-3i))(x-(1-i))(x-(2+3i))

f(x)=k((x-1)+i)((x-2)-3i)((x-1)-i)((x-2)+3i)

f(x)=k((x-1)+i)((x-1)-i)((x-2)-3i)((x-2)+3i)

f(x)=k((x-1)²-i²)((x-2)²-(3i)²)

f(x)=k((x-1)²-(-1))((x-2)²-(-9))

f(x)=k((x-1)²+1)((x-2)²+9)

f(x)=k((x²-2x+1)+1)((x²-4x+4)+9)

f(x)=k(x²-2x+2)(x²-4x+13)

f(x)=k(x⁴-6x³+23x²-34x+26)