Is it possible to transform any polynomial into this specific alternative notation?

For example, for the polynomial $$16-10x+x^2$$

There exists an alternative notation: $$f(x)=(x−2)(x−8)$$

I have seen it around quite a bit. Does it have a name? I would like to know more about it but can't research it without knowing the name.

I also wonder if and how it would be possible to take any univariate polynomial and transform it into this form. Is there an algorithm for that? I expect that I'd have to add a constant factor in case the highest degree factor is not $$1$$ like the example, but that would be okay.

• That's called factoring. It's typically taught in high school algebra courses in the 9th grade (age 14, in US). Commented Oct 23, 2022 at 23:17
• Um.... what is the nature of this "alternate notation"? Is it a matter of factoring? Must coefficients be rational? most they be real? In general not all polynomials can be factored. Not all polynomials have real roots. Commented Oct 23, 2022 at 23:19
• Re "Does it have a name?" Yes: factored form. Commented Oct 24, 2022 at 1:06

To elaborate what the commenters said, you might be searching for the term "factoring". However, it is still unclear how your alternative notation behaves. Not every polynomial can be factored into first-degree factors (or factors of the form $$x - a$$). For example, try $$x^3 - 1$$. It factors out into $$(x - 1)(x^2 + x + 1)$$ but $$x^2 + x + 1$$ is not a first-degree factor but a second-degree factor.
The Abel-Ruffini theorem states that not all fifth-degree polynomials and higher can be solved by radicals. This means that, using only the arithmetic operations and radicals, one can't just solve a particular polynomial. Taking the example from the linked article, the simplest non-solvable fifth-degree polynomial is $$x^5 - x - 1$$.
Moreover, requiring the roots of the polynomial to be real numbers may leave us no choice but to use the polynomial as itself. Try factoring $$x^2 + 1$$ and you will see that this polynomial can't be factorable over the real numbers.