Complex/surd conjugates (1 Viewer)

[chanandlerbong]

My question:
I know that if the coefficients of P(x) is real, that means the complex conjugate is real if there is a complex root.
However, what if there is a surd root? e.g root5, would the conjugate -root5 also be a root?

 

[KingOfActing]

My question:
I know that if the coefficients of P(x) is real, that means the complex conjugate is real if there is a complex root.
However, what if there is a surd root? e.g root5, would the conjugate -root5 also be a root?

I'm not quite sure I understand what you're asking. If we're given some polynomial P with real coefficients, and we know , then by the complex conjugate root theorem, we also know .

On the other hand, by the irrational conjugate root theorem, if we are given a polynomial P with rational coeffecients, and then we also know , given that a, b and c are all rational, and c is not a perfect square.

 

[chanandlerbong]

When you say rational coefficients, you mean where numbers can be expressed as a fraction right? Also, I have another question:

P(z) = z^4 + (1-2i)z^2 -2i.
i is a root of this equation. Explain why -i is also a root.

I can't talk about the conjugate root theorem right? So how do I explain this? Do I talk about how P(z) has even powers, therefore it's an even function? So -i is also a root? I'm a little confused

 

[Paradoxica]

When you say rational coefficients, you mean where numbers can be expressed as a fraction right? Also, I have another question:

P(z) = z^4 + (1-2i)z^2 -2i.
i is a root of this equation. Explain why -i is also a root.

I can't talk about the conjugate root theorem right? So how do I explain this? Do I talk about how P(z) has even powers, therefore it's an even function? So -i is also a root? I'm a little confused

Yes, even powers, because the 2n-th power of i is equal to the 2n-th power of -i.

In an even polynomial, if a is a root, then -a is a root, without regards to complex/real roots.

That or direct substitution.

 

[InteGrand]

When you say rational coefficients, you mean where numbers can be expressed as a fraction right? Also, I have another question:

P(z) = z^4 + (1-2i)z^2 -2i.
i is a root of this equation. Explain why -i is also a root.

I can't talk about the conjugate root theorem right? So how do I explain this? Do I talk about how P(z) has even powers, therefore it's an even function? So -i is also a root? I'm a little confused

Edit: said above