1.7 Continuing Example 1.3

In Example 1.3 we had $g(x)=x^{3}+x^{2}+3x-5$. Clearly $g(1)=1+1+3-5=0$ so $g(x)$ is divisible by ${(x-1).}$ Using polynomial division again, we have $g(x)={(x-1)(x^{2}+2x+5).}$

We have to determine whether $x^{2}+2x+5$ is irreducible. To do this we complete the square:

is irreducible. Alternatively we use the formula for the roots of a quadratic polynomial: the roots are

which are complex, so $x^{2}+2x+5$ is irreducible over ${\mathbb{R}}$.