Find a polynomial function of least degree having only real​ coefficients, a leading coefficient of​ 1, and zeros of 4 and 3+i

Answer 1

When one of the zeros is a complex number, it comes in conjugate pairs. i = √(-1)

x = 3 + i

x - (3 + i) = 0

x = 3 - i

x - (3 - i) = 0

[x - (3 + i)]*[x - (3 - i)] = x² - x(3 - i) - x(3 + i) + (3 + i)(3 - i)

= x² - 3x + xi - 3x - xi + 9 - (i)²

= x² - 3x + xi - 3x - xi + 9 + 1

= x² - 6x + 10

x = 4

x - 4 = 0

(x - 4)*(x² - 6x + 10) = x³ - 6x² + 10x - 4x² + 24x - 40 = x³ - 10x² + 34x - 40