Question

Given a degree of 3, zeros at -1 and 2 + √(5i) and a solution point at f(-2) = 42 I am supposed to find a) the polynomial in completly factored form and b) the polynomial in expanded form.

It's pretty easy until the solution in the book has me stumped.

I get all the factors including the conjugate for 2 + √(5i)
(x + 1)(x - 2 - √(5i))(x - 2 + √(5i))
This is where I get stuck. When I multiply the complex factors I get
x² - 4x + 4 - 5i
Because -√(5i) × √(5i) = -(√(5i))² = -5i?

But the book gets
x² - 4x + 4 + 5
If I can understand why this is then I can easily finish the problem.

Answer 1

Surely your text means \(\sqrt 5i\) as in the square root of 5 multiplied with \(i\), and not \(\sqrt{5i}\) as in the square root of the quantity \(5i\).

Answer 2

As for your work assuming \(\sqrt{5i}\), your expansion of the terms containing complex roots is correct.

Answer 3

Nope, I was given the square root of 5i. This is why I am confused on how the book got 5 instead of -sqrt of 5i

Answer 4

Lol! Now I feel like an idiot. I went back and double checked the problem to make sure it wasn't as you said and upon closer inspection it was. It really does look like it is underneath the sqaure root especially the problem after it, but you are correct. Sorry for wasting your time.

Answer 5

No problem. Some texts avoid that problem by writing complex numbers in the form \(a+ib\), so if \(b\) happens to be some complicated expression (relative to a real number, for instance), then there's little room for ambiguity.

Answer 6

Do you still need help ?