High School Roots of Polynomials: Understanding Mathematical Methods

[Rishabh Narula]

I was reading this book - " mathematical methods for physics and engineering"
in it in chapter 1 its says
"F(x) = A(x - α1)(x - α2) · · · (x - αr),"
this makes sense to me but then it also said

We next note that the condition f(αk) = 0 for k = 1, 2, . . . , r, could also be met
if (1.8) were replaced by
F(x) = A(x - α1)^m1(x - α2)^m2 · · · (x - αr)^mr

this confused me...how can you can you just raise each factor to powers m1,m2,...mr etc.How does the function still remain same?

 

[HallsofIvy]

The function doesn't "remain the same" and this doesn't say it does. It just says that the two functions have the same zeros. Both (x- a_r) and (x- a_r)^n are 0 if and only if x= a_r.