If a polynomial function with rational coefficients has the zeros 2, #-2+sqrt10#, what are the additional zeros?

1 Answer
Jan 15, 2017

Answer:

#-2-sqrt(10)# must also be a zero and the function will be a multiple (scalar or polynomial) of:

#f(x) = x^3+2x^2-14x+12#

Explanation:

Given that #2# and #-2+sqrt(10)# are zeros, the rational conjugate #-2-sqrt(10)# must also be a zero.

The polynomial function will be a multiple (scalar or polynomial of this #f(x)#:

#f(x) = (x-2)(x+2-sqrt(10))(x+2+sqrt(10))#

#color(white)(f(x)) = (x-2)((x+2)-sqrt(10))((x+2)+sqrt(10))#

#color(white)(f(x)) = (x-2)((x+2)^2-(sqrt(10))^2)#

#color(white)(f(x)) = (x-2)(x^2+4x+4-10)#

#color(white)(f(x)) = (x-2)(x^2+4x-6)#

#color(white)(f(x)) = x^3+2x^2-14x+12#