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I am dealing with an issue for which I do not find answer on the Internet. When I factorize a polynomial, I can get this structure:

$$ (x-a)(x-b)(x-c)^2 $$

But sometimes I have seen others like: $$ k(x-a)(x-b)(x-c)^2 $$

Where $k$ is any real number. What does it mean? I know it is related with the coefficient of the root with the highest algebraic multiplicity but I don't get to understand it.

It could be $\frac{1}{3}(x-1)(x-2)$ but never $(x-1)(x-2)+\frac{1}{3}$ isn't it? Why? Thank you a lot :)

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up vote 2 down vote accepted

$k$ is the leading coefficient, the coefficient of the highest power of the variable. That is, if you expand out $k (x - a)(x-b)(x-c)^2$ you'll get $k x^4 + $ (terms in lower powers of $x$). It has nothing to do with the roots.

On the other hand, $(x-1)(x-2)+\frac{1}{3}$ is not a factorization. To factorize something means to write it as a product of things, not as a product plus something else.

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Thank you a lot. Simple and complete! – Mark Tower Dec 14 '12 at 8:48

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