How do you find a polynomial function of degree 6 with -1 as a zero of multiplicity 3, 0 as a zero of multiplicity 2, and 1 as a zero of multiplicity 1?

1 Answer
Nov 5, 2016

Answer:

#f(x) = x^6+2x^5-2x^3-x^2#

Explanation:

Each zero (e.g. #a#) corresponds to a linear factor (e.g. #(x-a)#).

Multiplicity corresponds to a repetition of that factor.

So in our example, the following polynomial fits the criteria:

#f(x) = (x-(-1))^3(x-0)^2(x-1)#

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

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

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

#color(white)(f(x)) = x^2(x^4+2x^3-2x-1)#

#color(white)(f(x)) = x^6+2x^5-2x^3-x^2#

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