How to express that roots of a polynomial are real

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[Alex Gu]

How can I express that the roots of a polynomial are all real? For example, that x^3-7x^2+14x-8 has all real roots.

Would the easiest way be like "if a complex number a satisfies ..., then a must be real"?

[Aaron Liu]

Here are some things that come to mind:

• You could say that the polynomial has 3 (distinct) (real) roots
• You could say that "if a complex number a satisfies ..., then a must be real"
• You could say that the polynomial splits over the real numbers

It really depends on what you are using this theorem for (if this is your end goal, then pick your favorite statement).

[Eric Wieser]

You could also show that the real roots are the preimage of the complex roots under the coercion, but that's probably not very helpful