I'm preparing to my algebra exam. And I have problem and I have no idea how to solve it.
Given polynomial $$x^4+4x^3+4x^2+1.$$ The task is find expansion of the polynomial as a product of irreducible polynomials in $\mathbb{R}$.
I will be happy if you show me the way how to solve such problems
On
Hint:$$x^4+4x^3+4x^2+1=(x+1)^4-2(x+1)^2+2$$ $$(x+1)^4-2(x+1)^2+2=((x+1)^2-1)^2+1=((x+1)^2-1+i)((x+1)^2-1-i)$$ This will allow you to get the roots of the equation $x^4+4x^3+4x^2+1=0$. By multiplying the monomials with conjugate roots you will get the real quadratic factors.
To see why this always works:
It is a fact that if $P$ is a polynomial such that $P(z)=0$, then $P(\bar z)=0$.
Finally:
$$(x-(a+bi))(x-(a-bi))=x^2-2ax+a^2+b^2$$
Which is a real quadratic polynomial.
It suffices to show that $f(x)=x^4+4x^3+4x^2+1$ has no linear factors over $\mathbb{Z}_3$: $f(0)=f(1)=1$ and $f(2)=2$, so $f(x)$ has no linear factors. Then $f(x)$ must factor to two quadratic polynomials: $$f(x)=(ax^2+bx+c)(ux^2+vx+w)$$ We then have that $au=1$. Multiplying the first polynomial by $u$ and the second by $a$, we may assume that $a=u=1$. Equating the coefficients of the powers of $x$, we have $$ \begin{eqnarray*} 4&=&v+b\\ 4&=&w+c+bv\\ 0&=&bw+cv\\ 1&=&cw \end{eqnarray*} $$ Some algebra shows that $$\begin{eqnarray*} b&=& 2+\sqrt{2 \left(1+\sqrt{2}\right)}\\c&=& 1+\sqrt{2}+\sqrt{2 \left(1+\sqrt{2}\right)} \\ v&=& 2-\sqrt{2 \left(1+\sqrt{2}\right)}\\ w&=& 1+\sqrt{2}-\sqrt{2 \left(1+\sqrt{2}\right)} \end{eqnarray*} $$
So, letting $\alpha=1+\sqrt{2}$, we see that $x^4+4x^3+4x^2+1$ factors to $$\left(x^2+\left(2+\sqrt{2\alpha}\right)x+\alpha+\sqrt{2\alpha}\right)\left(x^2+\left(2-\sqrt{2\alpha}\right)x+\alpha-\sqrt{2\alpha}\right).$$