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I'm studying AS maths, and am trying to connect my thoughts around polynomials and the factor and remainder theorems.

I understand the factor theorem and its application: it helps me find roots of a polynomial.

The remainder theorem is also easy to understand, until I ask myself what the remainder is useful for. Why do I need it?

Here's a concrete example:

When $x^3 + 3x + k$ is divided by $x - 1$ the remainder is 6. Find the value of $k$.

So if I evaluate $f(1)$ for the polynomial, I'm given a constant, thus:

$f(1) = 6$ therefore $6 = 1^3 + 3(1) + k$ which when rearranged equals $2 = k$. So in this example it's easy to understand why I'd use it.

What else is the remainder useful for? Or is it just to find constants?

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In abstract algebra one of the earliest most useful algorithms is the division algorithm which extends nicely to factoring polynomials. Results of polynomial division in this section of mathematics will help determine multiplicative inverses in polynomial fields. One can also use synthetic division to evaluate polynomials at values of its domain.

$\textbf{Additional}$: Here is another use for polynomial division.

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