What is the polynomial function that has the zeros: 4-i and sqrt(10).?

Answer 1

Square root and complex zeros always come in pairs and when r is a zero x-r is a factor of the polynomial so we can construct your polynomial by multiplying:

p(x) = (x-sqrt10)(x+sqrt10)(x-4+i)(x+ 4 - i)

Pick two parenthesis at a time, multiply, and then multiply the results by each remaining parenthesis, using the distributive property of course.

Answer 2

(x^2-10)(x-4-i)(x-4+i)
(x^2-10)((x-4)^2+1)
(x^2-10)(x^2-8x+17)
x^4-8x^3+17x^2-10x^2+80x-170
x^4-8x^3+7x^2+80x-170