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## Main Question or Discussion Point

Hello everyone,

Going through calculus study, there is a vague point regarding polynomials I'd like to make clear.

Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##. If I understand correctly, ##h(x)=(x-a)g(x)## for all ##x≠a## and is undefined at ##x=a##.

Now, although ##h(a)## is undefined, it is quite clearly divisible by ##x-a##. The book I'm using seems to treat this as a polynomial, despite the fact that it has point where it is undefined. I'm guessing it does that by assuming that ##h(a)=0## or something of the sort, but I'm not sure, which leads my to my questions:

1. For a polynomial ##f##, are the statements that ##f(a)=0##, that ##f## has a root at ##a## and that that ##f## is divisible by ##x-a## identical?

2. Does dividing 2 polynomials (without a remainder) result in a polynomial? If so, can that polynomial have roots at points where the denominator of the division was 0?

Thanks in advance to all the helpers

Going through calculus study, there is a vague point regarding polynomials I'd like to make clear.

Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##. If I understand correctly, ##h(x)=(x-a)g(x)## for all ##x≠a## and is undefined at ##x=a##.

Now, although ##h(a)## is undefined, it is quite clearly divisible by ##x-a##. The book I'm using seems to treat this as a polynomial, despite the fact that it has point where it is undefined. I'm guessing it does that by assuming that ##h(a)=0## or something of the sort, but I'm not sure, which leads my to my questions:

1. For a polynomial ##f##, are the statements that ##f(a)=0##, that ##f## has a root at ##a## and that that ##f## is divisible by ##x-a## identical?

2. Does dividing 2 polynomials (without a remainder) result in a polynomial? If so, can that polynomial have roots at points where the denominator of the division was 0?

Thanks in advance to all the helpers