How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 4, 4, 2+i?

1 Answer
Sep 17, 2017

Answer:

#f(x) = x^4-12x^3+53x^2-104x+80#

Explanation:

Assuming you want a polynomial with real coefficients, the complex conjugate #2-i# must also be a zero and the monic polynomial of least degree is:

#f(x) = (x-4)(x-4)(x-(2+i))(x-(2-i))#

#color(white)(f(x)) = (x^2-8x+16)((x-2)+i)((x-2)-i)#

#color(white)(f(x)) = (x^2-8x+16)((x-2)^2-i^2)#

#color(white)(f(x)) = (x^2-8x+16)(x^2-4x+4+1)#

#color(white)(f(x)) = (x^2-8x+16)(x^2-4x+5)#

#color(white)(f(x)) = x^4-12x^3+53x^2-104x+80#

Any polynomial in #x# with these zeros will be a multiple (scalar or polynomial) of this #f(x)#.