How do you determine an equation of a polynomial function with zeros at x=2,-2,1 and y-intercept of 24?

1 Answer
Alan P.
Feb 18, 2016

#y=x^4+5x^3-10x^2-20x+24#

Explanation:

If #{2,-2,1}# are all zeros of the polynomial
then
the polynomial must contain the factors:
#color(white)("XXX")(x-2)(x+2)(x-1)#

So
#color(white)("XXX")y=(x-2)(x+2)(x-1)xxa# for some additional factor #a#

#bary = (x-2)(x+2)(x-1)=x^3-x^2-4x+4#
which is equal to #4# when #x=0#
So if #y=baryxxa# is to be equal to #24# when #x=0#
then #a# must have the value #6# when #x=0#

The most obvious (but not only) value for #a# is
#color(white)("XXX")(x+6)#
in which case
#color(white)("XXX")y=(x-2)(x+2)(x-1)(x+6)#
#color(white)("XXXX")=x^4+5x^3-10x^2-20x+24#

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