Sam S. answered • 10/21/20
Friendly, knowledgeable, and a passion for teaching.
Hi there! Great question! This problems seems really intimidating and I was even scared by it at first, until I realized that it has a trick that can be used in all sorts of problems like this. :) The key to this problem is that every polynomial must be divisible by its zeroes. Let's say a polynomial, P(x), has a zero at x=4. This then means that P(x)/(x-4) will yield another polynomial, one that has one degree less than the original. If a root has a multiplicity of more than 1 that means the original polynomial is divisible by the root to the power of the multiplicity. So if x=4 had a multiplicity of 2 then P(x)/((x-4)^2) would still be a polynomial, however since it is being divided twice it will have a degree of two less than the original instead of one. The trick here is that we can use this property to work backwards to solve this problem! Instead of dividing out our starting poylnomial by the zeroes we multiply them together to find the polynomial. This means that P(x)=(x^2)((x-4)^2)(x+2). We can then use distributive property of multiplication to expand this out and see that P(x)=x^5+-6x^4+32x^2, giving us our answer! :)
