Question

Using the given zero, find one other zero of f(x). Explain the process you used to find your solution.

1 - 6i is a zero of f(x) = x^4 - 2x^3 + 38x^2 - 2x + 37.

Answer 1

What do you know about polynomials with real coefficients and complex roots?

Answer 2

That is the lesson I am doing now.

Answer 3

Okay. Do you see any complex coefficients in your polynomial?

Answer 4

If not, that means you have only real coefficients. If you have only real coefficients, that means either you have only real roots, or any complex roots come in conjugate pairs. Do you know what that means?

Answer 5

no i don't see any...?

Answer 6

a complex number can be written in the form \(a + bi\) where \(i^2=-1\). \(a\) is the real part, \(bi\) is the imaginary part. The conjugate of \(a+bi\) is \(a-bi\). If you multiply a conjugate pair together, something magical happens:

\[(a+bi)(a-bi) = a^2-abi + abi -b^2i^2 = a^2-b^2i^2\]but \(i^2=-1\) so that becomes\[a^2-b^2(-1)= a^2+b^2\]Look, no more complex part!

Answer 7

They must occur in conjugate pairs so the "imaginary" coefficients vanish!

Answer 8

The result is that if you have only real coefficients, any complex roots must come in conjugate pairs (or you'd have an \(i\) somewhere in the equation's coefficients). That means that you know one other root — the conjugate of the complex root you have.

Answer 9

So, what is the conjugate of \(1-6i\) — that is an answer to your problem.

You could use this knowledge to find all of the roots of this equation. Multiply out \[(x-(1-6i))(x-(1+6i))\]and divide the polynomial by the result. The remaining polynomial will have the same roots as the original polynomial, less the roots we just identified. I think if you do the division correctly you'll see how to factor the result and find the remaining roots (which are also a complex conjugate pair).

Answer 10

Can you just tell me the answer and then I can work it back or try until I get it. That would be what works best for me.

Answer 11

Okay, I told you the roots come in conjugate pairs. I described a conjugate pair. I pointed out that the root you have is one half of a conjugate pair. Al you have to do is find the conjugate of the root you already have. If your root looks like a+bi, it's conjugate is a-bi. If it looks like a-bi, the conjugate is a+bi.

Answer 12

I understand that.

Answer 13

Your root is 1-6i so the conjugate is...

Answer 14

1+6i

Answer 15

Now how do I explain what I did?

Answer 16

You took advantage of the fact that complex roots in polynomials with only real coefficients always come in conjugate pairs.

Answer 17

To construct the conjugate root, you changed the sign of the imaginary part.