Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 2, 5−i

Question

I need help with this problem i don't understand it.

Answer 1

Since the polynomial has rational coefficients, complex zeros occur in conjugate pairs.  So, since 5 - i is a zero, so is 5 + i.

 

Also, if c is a zero of a polynomial function y = f(x), then x - c is a factor.

 

f(x) = (x - 2)[x - (5-i)][x - (5+i)]

 

      = (x - 2)[(x-5) + i][(x-5) - i)

 

      = (x - 2)[(x-5)² - i²]

 

      = (x - 2)(x² - 10x + 25 + 1)

 

      = (x - 2)(x² - 10x + 26)

 

      = x³ - 10x² + 26x - 2x² + 20x - 52  = x³ - 12x² + 46x - 52