Three tricky questions that need solving..

Hello MHF!

I have three questions that which are throwing me for a loop. While I understand some of the principals behind them, the questions themselves defy my ability to work out.

Q1:

Simplify:

$\displaystyle \frac{1}{2-i}$

(I have no idea where to begin to work this one out, or even if I have to...)

Q2:Find a quadratic equation with the roots (4+i) and (4-i).

(I have no idea where to start with this one either.......)

Q3:Use the factor theorem to determine whether (x-3) is a factor of $\displaystyle f(x)=x^4+12x^3+6x+27$

(I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

Many thanks in advance!

bad type setting on my part.

Quote:

Originally Posted by

**e^(i*pi)** You'd be dividing f(x) by (x-3). I never learnt synthetic division so I can't tell you how you'd work it out

Well....I remember that when doing synthetic division the non-variable aspect of the divisor is used to "divide" the coefficients, but I can't remember if I am to use 3 or -3 in the place of the 'r'.

__3|__

1 12 0 6 27

-3 -27 81 -261

_________________

1 9 -27 87 |-234

or, __3|__

1 12 0 6 27

3 45 135 423

__________________

1 15 45 141 | 450

I know that the second version's remainder coincides with the function as worked with x=3. Therefore I am guessing that the second version is correct from what I understand of synthetic division and the remainder theorem, but I just want to make sure.

Because I worked the function as f(3) and it does not equal 0, I *think* that means that (x-3) is *not* a factor, right?