Simplest way to get the real solutions for $ x^{4}-2x+1=0$?

Question

What is the simplest way to get the real solutions for this equation?

$$ x^{4}-2x+1=0 $$

I can do it and also ask for a step-by-step solution to Wolfram Alpha, but I was wondering if someone has a simpler way...

Answer 1

Rational root test tells us that if there is a rational root $p/q$, then $p$ divides free coefficient $a_0 = 1$ and $q$ divides leading coefficient $a_4 = 1$. Thus, the only possible rational roots are $\pm 1$. Quick check confirms that $1$ is a root, but $-1$ is not. Thus, $x^4-2x+1$ is divisible by $x-1$ and the corresponding factorization is $$ x^4-2x+1 = (x-1)(x^3+x^2+x-1) $$

Unfortunately, $x^3+x^2+x-1$ doesn't have any rational roots (so, it is in fact irreducible over $\mathbb Q$) and to find the roots, you can follow Wiki's article on cubic function.

Answer 2

Notice that the sum of coefficients is 0, so $1$ is a solution. Divide by $x-1$ and simplify.

Answer 3

Note that sum of coefficients are $0$ so $x=1$ is a real root. Then divide by $x-1$ and focus on the quotient polynomial for another real root.