Prove function is greater than or equal to 0

Question

Let $f(x)=x^4-2x^3+3x^2-2x+1$. Prove that $f(x) \ge 0$ .

My thought is I need factor the function into sum or difference of perfect squares to show it's always non-negative. Any suggestions?

Answer 1

It's just $$(x^2-x+1)^2>0$$ Also, we can use your idea: $$f(x)=x^4-2x^3+x^2+x^2+x^2-2x+1=x^2(x-1)^2+x^2+(x-1)^2>0.$$