King_Ghidra

05-06-2006, 16:07:54

Number 1.

Let c = a+bi and c* = a-bi be two conjugate roots of a polynomial p(x),

Then the polynomial factors

p(x) = A (x-c)(x-c*)(x-r1)...(x-rk),

where r1, r2,...,rk are the other roots. Define

f(x) = (x-c)(x-c*) = x^2 - 2ax + (a^2+b^2),

g(x) = A (x-r1)...(x-rk),

so that p(x) = f(x)g(x). Finally, let K = g(a). Then for x near a on

the number line,

p(x) ~ f(x)g(a) = K f(x). (*)

In other words, for x near a, p(x) looks approximately like K f(x), which

is a quadratic. Since the apex of this quadratic is at x=a, it has a

wiggle (hump) near x=a. Thus p(x) ~ K f(x) also should have a hump

near x=a.

Let c = a+bi and c* = a-bi be two conjugate roots of a polynomial p(x),

Then the polynomial factors

p(x) = A (x-c)(x-c*)(x-r1)...(x-rk),

where r1, r2,...,rk are the other roots. Define

f(x) = (x-c)(x-c*) = x^2 - 2ax + (a^2+b^2),

g(x) = A (x-r1)...(x-rk),

so that p(x) = f(x)g(x). Finally, let K = g(a). Then for x near a on

the number line,

p(x) ~ f(x)g(a) = K f(x). (*)

In other words, for x near a, p(x) looks approximately like K f(x), which

is a quadratic. Since the apex of this quadratic is at x=a, it has a

wiggle (hump) near x=a. Thus p(x) ~ K f(x) also should have a hump

near x=a.