WitCH 119: Poly Want a Cracker?

Last one. These are excerpts from the final section of Nelson‘s complex numbers chapter. Similar to the previous WitCHes, I’ve tried to not be manipulative material in selecting the material except, of course, in selecting the worst bits: the worked examples not indicated are standard, and in general the working is tedious but ok; a monotonous but essentially correct proof of the conjugate root theorem is included in the text.

7 Replies to “WitCH 119: Poly Want a Cracker?”

1. Joe says:

Box 1:

*Non-real* roots of real quadratic equations must be complex conjugates. (They make this mistake throughout, and it would be cumbersome to repeat it every instance it occurs, so I’ll just leave it here.)

What does it mean by a z-term? Define the polynomial in terms of z first. And “using the roots of complex numbers”—what the hell do they mean?

Box 6:

The second sentence is incorrect even by their misunderstanding of the definition of complex. If you take “complex” to mean “non-real”, has no complex roots, but it has complex coefficients. If you take “complex” to mean what it actually means, all roots of polynomials with complex coefficients are necessarily complex.

1. marty says:

Thanks, Joe. Can you tell me what Nelson means by “complex cubic/quartic equation”?

1. Joe says:

From the looks of it they mean a cubic/quartic equation with complex coefficients.

1. marty says:

Are you using “complex coefficients” in the Nelson sense or in the English sense?

1. Joe says:

The Nelson sense.

1. marty says:

Ok thanks. Look more carefully.

2. Yu says:

I think Nelson means:
In a second degree polynomial, you have a z^2 term, a z term and a constant.