How do you find a polynomial function that has the given zeros. 0, –3?

1 Answer
Nov 5, 2015

Answer:

#x^2+3x#

Explanation:

We know that if a product #a*b=0#, then either #a=0# or #b=0# (or both). So, if we want a polynomial to have certain zeroes, in this case #0# and #-3#, we can multiply to polynomials that have those zeroes.

It's clear that the choice is not unique, but we usually choose the easiest ones: if we want #x_0# to be a zero of the polynomial, #(x-x_0)# is surely a good option, since the result in #x_0# gives #x_0-x_0=0#.

So, if we want #0# to be a solution, our polynomial would be #(x-0)=x#. As for #-3#, the same steps lead us to #(x-(x-3))=(x+3)#

Now we have a polynomial with a root in zero (namely #x#), and a polynomial with a root in #-3# (namely #x+3#). If we multiply them, we have the desired polynomial:

#x(x+3)=0# if #x=0# or #x+3=0#, i.e. #x=-3#.

To write it as an explicit polynomial, simply expand the expression:

#x(x+3)=x^2+3x#