1. ## Arghhhh Argument Principle

I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

Anyway, I'm not sure on this question:

By sketching the graph of $\displaystyle y = f(x)$ show that

$\displaystyle f(z) \equiv z^4 + 4z + 1 = 0$

has two negative real roots and no postitive real roots. Show also that $\displaystyle f(z)=0$ has no purely imaginary roots.

Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.

2. Originally Posted by Jason Bourne
I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

Anyway, I'm not sure on this question:

By sketching the graph of $\displaystyle y = f(x)$ show that

$\displaystyle f(z) \equiv z^4 + 4z + 1 = 0$

has two negative real roots and no postitive real roots. Show also that $\displaystyle f(z)=0$ has no purely imaginary roots.

Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.
Well, sketch the graph of $\displaystyle y = x^4 + 4x + 1$. You don't have to have an art degree, just get some graph paper and plug in some values of x and connect the dots with a smooth curve.

You should be able to see that there are two negative and no positive roots.

I don't know how to show that the other two roots are not pure imaginary by looking at the graph.

-Dan

3. Originally Posted by Jason Bourne
I get really annoyed when questions ask you to sketch stuff, like am doing maths and i'm supposed to be able to do art!

Anyway, I'm not sure on this question:

By sketching the graph of $\displaystyle y = f(x)$ show that

$\displaystyle f(z) \equiv z^4 + 4z + 1 = 0$

has two negative real roots and no postitive real roots. Show also that $\displaystyle f(z)=0$ has no purely imaginary roots.

Using the argument principle, determine in which quadrants of the complex plane the two complex roots lie.
Suppose that $\displaystyle z=ai$ for $\displaystyle a\not = 0$.
Then $\displaystyle f(ai) = a^4 +4ai+1=0$ by equation real and imaginary parts we get $\displaystyle (a^4+1)=0\mbox{ and }4a=0$ which is impossible.

What does this have to do with the principle of the argument?