Why is x^2 + 1 an Irreducible Polynomial?

[mathdad]

Why is x^2 + 1 irreducible?

 

[Joppy]

Why is x^2 + 1 irreducible?

What do you think? If you factor this expression, what do you get? Moreover, what could be said about those factors in relation to the real numbers?

 

[mathdad]

Complex numbers?

 

[Joppy]

Indeed!

In this case we could say that the expression is irreducible over the reals, but reducible over the complex numbers. But I might let someone else chime into give a more formal approach. :)

 

[mathdad]

Cool.

 

[mathdad]

What about x^2 + x + 1?

 

[MarkFL]

What about x^2 + x + 1?

What does the discriminant tell you about the roots of this quadratic polynomial?

 

[mathdad]

b^2 - 4ac

1^2 - 4(1)(1)

1 - 4

-3

b^2 - 4ac < 0

This means no real solutions.