1. Roots of a polynominalp(x)

Hey, I was wondering if anybody would be able to explain how to arrive to (a). It's in my notes but I don't seem how they got there?

Find the roots of the following polynominal.

p(x) = x^4 - x^3 + 4x^2 + 3x + 5

Info: b = 1 + 2i is a root.

Sp b(bar) = 1-2i should also be a root.

Try to divide p(x) by (x-(1+2i)) (x-(1-2i))
(a) = x^2 - 2x +5

Thanks :]

2. Originally Posted by schrei
Hey, I was wondering if anybody would be able to explain how to arrive to (a). It's in my notes but I don't seem how they got there?

Find the roots of the following polynominal.

p(x) = x^4 - x^3 + 4x^2 + 3x + 5

Info: b = 1 + 2i is a root.

Sp b(bar) = 1-2i should also be a root.

Try to divide p(x) by (x-(1+2i)) (x-(1-2i))
(a) = x^2 - 2x +5
Multiply out the outer brackets: $(x-(1+2i)) (x-(1-2i)) = x^2 -(1+2i+1-2i)x + (1+2i)(1-2i) = x^2 - 2x + (1+2i)(1-2i)
$
. Now multiply out the remaining brackets and you should see the required result.

3. Just a little simpler, I think.

(x-(1+2i)) (x-(1-2i))= [(x-1)+ 2i][(x-1)- 2i], a product of "sum and difference". Their product, then, is $(x-1)^2- (2i)^2= x^2- 2x+ 1+ 4= x^2- 2x+ 5$.