Factor $f(x) = x^5 - 2x^4 - x^3 + 2x^2 -2x + 4 \in \mathbb{Q}[x]$ into a product of irreducible polynomials in $\mathbb{Q},\mathbb{R},\mathbb{C}$

Question

I need help in the problem factor a polynomial into a product of irreducible polynomials.

Problem: Factor this polynomial $f(x) = x^5 - 2x^4 - x^3 + 2x^2 -2x + 4 \in \mathbb{Q}[x]$ into a product of irreducible polynomials in $\mathbb{Q},\mathbb{R},\mathbb{C}$.

Answer 1

Hint: Your polynomial is equal to $$ \left( x-2 \right) \left( {x}^{2}-2 \right) \left( {x}^{2}+1 \right) $$

Answer 2

Hint:

• Over $\mathbb{Q}$ we have $$f(x)=(x-2)(x^2-2)(x^2+1)$$
• Over $\mathbb{\mathbb{R}}$ we have $$f(x)=(x-2)(x-\sqrt{2})(x+\sqrt{2})(x^2+1)$$
• Over $\mathbb{C}$ we have $$f(x)=(x-2)(x-\sqrt{2})(x+\sqrt{2})(x-i)(x+i)$$

Can you show that each of the factors is irreducible over the respective field?