How to compute the roots of $x^3 - x^2 - 4x + 4$ = $0$

Question

I am wondering whether there is a simple way to find the roots of $x^3 - x^2 - 4x + 4$ = $0$ by algebraic manipulation

I will accept if this is not a trivial equation to compute the roots of

Any help is appreciated!

Answer 1

You can factor the polynomial as follows: $$x^2(x-1)-4(x-1)=0,$$$$(x^2-4)(x-1)=0.$$ The first factor can be factored further using difference of two squares, and then you easily get the three roots.

Answer 2

hint: it separates out into $(x^3-x^2)-(4x-4)$. Try out rational roots if you want how to do it without, uh, "seeing" it, though.

Answer 3

Have you tried using the rational root theorem?

Answer 4

$$\begin{align}x^3-x^2-4x+4\\=x^3-4x-x^2+4&\\=x(x^2-4)-(x^2-4)&\\=(x-1)(x^2-4)&\\=(x-1)(x-2)(x+2)\end{align}$$

Answer 5

Hint: $$x^3+x^2-4x-4=x^2(x-1)-4(x-1)=(x^2-4)(x-1)=(x+2)(x-2)(x-1)$$