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What is the following "form" on the right side of the bi-implication called?

$$-k^3 + 10k^2 -31k + 30 \leftrightarrow -(k+5)(k+3)(k+2)$$

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    you can say: "factored form" – 2017-01-25
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    is it possible to "transform" every polynomial function with form "ax^3+bx^2+cx+d" into a factored form? Edit: From wiki: "The polynomial x2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b)." i think it also applies to higher polynoms :) Thansk Arnaldo – 2017-01-25
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    Look the Fundamental Theorem of Algebra: https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra – 2017-01-25

3 Answers 3

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The right side is the factorized form of the left.

The factorized form always exists if the field you're working in is algebraically closed. In particular, since the complex numbers are algebraically closed, every polynomial $p(x)$ can be factored into some constant times factors of the form $x-r_i$ where $r_i$ is a root of $p(x)$.

$$ p(x) = c(x-r_1)(x-r_2)\cdots(x-r_n) $$

Where $p(x)$ is of degree $n$. Moreover, this factorization is unique up to the ordering of the linear factors $x-r_i$.

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Its simply factorised form.

Yes you can factorised in most cases and in some cases you get imaginary values.

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Every polynomial can be factored as

$$a_1x^n + a_2x^{n-1} + ... a_n x + a_{n+1} = a_1(x-r_1)(x-r_2)\cdots(x-r_{n-1})(x-r_n) $$

where $r_1,r_2,...,r_n$ are the roots of the polynomial (they may be complex).