[Math] How to factor $x^3-3x^2+3x-1$

algebra-precalculusfactoringpolynomials

$$x^3-3x^2+3x-1?$$

I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I don't know how to get there. any help would be greatly appreciated. and also if you have any recommended web sites that help with higher order polynomial factoring that would be extremely helpful.

Best Answer

First, guess a root (this is the hard part). The so called "rational roots test" will be helpful here.

Eventually, you'll discover that $x=1$ is a root. This will imply that your polynomial has the form $$ \tag{1}(x-1)(ax^2+bx+c), $$ for some constants $a, b, c$.

To find those constants, you could do one of two things (and maybe more)

  1. perform the division $ x^3-3x^2+3x+1\over x-1$.
  2. expand (1) and set it equal to the original polynomial. Setting the coefficients of the two sides of this equation equal to each other will give you a system of equations that are solvable for $a$, $b$, and $c$.

Once you've figured out what $a,b$, and $c$ are, factor the quadratic.

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