How do you create a polynomial p which has zeros #c=1, c=3#, #c=-3# is a zero of multiplicity 2, the leading term is #-5x^3#?

1 Answer
Aug 6, 2017

Answer:

#-5x^3+5x^2+45x-45#

Explanation:

Note that #x# is a zero of a polynomial if and only if #(x-c)# is a factor of that polynomial.

So in order to have zeros #1#, #3# and #-3#, our polynomial must be a multiple of:

#(x-1)(x-3)(x+3) = (x-1)(x^2-9) = x^3-x^2-9x+9#

In order that its leading term be #-5x^3#, we just need to multiply it by #-5# to get:

#-5(x^3-x^2-9x+9) = -5x^3+5x^2+45x-45#