The two theorems are similar, but refer to different things

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The two theorems are similar, but refer to different things. See explanation.

The remainder theorem tells us that for any polynomial ##f(x)##, if you divide it by the binomial ##x-a##, the remainder is equal to the value of ##f(a)##.

The factor theorem tells us that if ##a## is a zero of a polynomial ##f(x)##, then ##(x-a)## is a factor of ##f(x)##, and vice-versa.

For example, let's consider the polynomial

##f(x) = x^2 - 2x + 1##

Using the remainder theorem

We can plug in ##3## into ##f(x)##.

##f(3) = 3^2 - 2(3) + 1## ##f(3) = 9 - 6 + 1## ##f(3) = 4##

Therefore, by the remainder theorem, the remainder when you divide ##x^2 - 2x + 1## by ##x-3## is ##4##.

You can also apply this in reverse. Divide ##x^2 - 2x + 1## by ##x-3##, and the remainder you get is the value of ##f(3)##.

Using the factor theorem

The quadratic polynomial ##f(x) = x^2 - 2x + 1## equals ##0## when ##x=1##. This tells us that ##(x-1)## is a factor of ##x^2 - 2x + 1##.

We can also apply the factor theorem in reverse:

We can factor ##x^2 - 2x + 1## into ##(x-1)^2##, therefore ##1## is a zero of ##f(x)##.

Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its .