How do you create a polynomial #p# which has zeros #x=+-3, x=6#, leading term is #7x^5#, and the point #(-3,0)# is a local minimum on the graph of #y=p(x)#?

1 Answer
Feb 19, 2018

Answer:

#f(x) = 7x^5+21x^4-315x^3-945x^2+2268x+6804#

Explanation:

Each zero #x=a# corresponds to a linear factor #(x-a)#

We need the zero at #(-3, 0)# to be of even multiplicity in order that it is also a local minimum or maximum.

Also, since the leading coefficient is positive and the point #(-3, 0)# is a local minimum, there must be another real zero less than #-3#.

So let's write:

#f(x) = 7(x+6)(x+3)(x+3)(x-3)(x-6)#

#color(white)(f(x)) = 7(x+3)(x^2-36)(x^2-9)#

#color(white)(f(x)) =7x^5+21x^4-315x^3-945x^2+2268x+6804#

graph{7x^5+21x^4-315x^3-945x^2+2268x+6804 [-10, 10, -12000, 10000]}