Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. 1, 2−√3

Answer 1

zeros are 1 and 2-sqr3

but the irrational zeros come in conjugate pairs, so a necessary 3rd zero is 2+sqr3

factors are (x-1) and x^2+4x+1

x-1 times x^2- 4x+ 1 = x^3 -5x^2+5x -1 = the least degree polynomial with the given zeros and leading coefficient = 1

f(x) = x^3 -5x^2 +5x -1

x = 2-sqr3 means a factor of x-2+sqr3

x = 2+sqr3 means a factor of x-2-sqr3

mutliply them to get x^2-4x+1

multply that by x-1 to get the cubic polynomial