How do you write a polynomial in standard form given the zeros 3, 1, 2, and-3?

1 Answer
Jun 29, 2018

Answer:

#f(x) = x^4-3x^3-7x^2+27x-18#

Explanation:

Using #x# as our variable name, each zero #a# corresponds to a factor #(x-a)#.

So we can write:

#f(x) = (x-3)(x-1)(x-2)(x+3)#

#color(white)(f(x)) = ((x-3)(x+3))((x-1)(x-2))#

#color(white)(f(x)) = (x^2-9)(x^2-3x+2)#

#color(white)(f(x)) = x^2(x^2-3x+2)-9(x^2-3x+2)#

#color(white)(f(x)) = (x^4-3x^3+2x^2)-(9x^2-27x+18)#

#color(white)(f(x)) = x^4-3x^3-7x^2+27x-18#

This is the simplest polynomial with the given zeros. Any polynomial in #x# with these zeros is a multiple (scalar or polynomial) of this #f(x)#.