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When I need to factorise the expression $x^2 + x - 12,$ I immediately recognise that it is a quadratic and that it factorises into $(x + 4)(x - 3).$ I know that I need to use the quadratic formula (or in this case work it out mentally).

A more obvious example is when finding the area of a rectangle. I know that I need to multiply the length by the width.

My question is, when is long division of polynomials required? Please give some specific examples.

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    What is "normal" algebra? – 2017-02-04
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    Hindu-Arabic numbers *are* polynomials so... – 2017-02-04
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    I can think of several answers to what I think might be your question. This is why the site insists on context. Please give us some. Then we won't have to fumble around guessing what you want. – 2017-02-04
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    I've updated the question. – 2017-02-04

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In factorizing some particular cubic expression like $x^3-13x+12$, we have to use long division to figure out the factors

And you may get $(x-1)(x^2-12)$ and furthermore $(x-1)(x-2\sqrt3)(x+2\sqrt3)$

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    But how did you get to that factorization? There are an infinite amount of possible roots, and guess and check is no better than the OP's strategy. – 2017-02-04
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    It seems to me that in this case, long division is not particularly useful in finding the root $x=1$ (though IIRC there is an algorithm for finding such a root that looks vaguely like long division, based on the Rational Root Theorem). However, once you know about that root (regardless of how you found it), long division lets you factor it out and obtain the other factor, $x^2-12,$ after which the rest of the factorization is easy. – 2017-02-04