# Math Help - Three tricky questions that need solving..

1. ## 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:
$\frac{1}{2-i}$
(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).

Q3:Use the factor theorem to determine whether (x-3) is a factor of $f(x)=x^4+12x^3+6x+27$
(I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

2. Hi there MathBlaster47

Originally Posted by MathBlaster47

Q1:
Simplify:
$\frac{1}{2-i}$
(I have no idea where to begin to work this one out, or even if I have to...)
Using the complex conjugate, expand the following

$\frac{1}{2-i}\times\frac{2+i}{2+i}$

Originally Posted by MathBlaster47

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

expand $(x-(4+i))(x-(4-i))$

Originally Posted by MathBlaster47

Q3:Use the factor theorem to determine whether (x-3) is a factor of $f(x)=x^4+12x^3+6x+27$
(I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

will be a factor if $f(3)=0$

now complete $f(3)=3^4+12\times 3^3+6\times 3+27$

is this zero?

3. Originally Posted by pickslides
Hi there MathBlaster47

Using the complex conjugate, expand the following

$\frac{1}{2-i}\times\frac{2+i}{2+i}$

expand $(x-(4+i))(x-(4-i))$

will be a factor if $f(3)=0$

now complete $f(3)=3^4+12\times 3^3+6\times 3+27$

is this zero?
I Thank you heartily Pickslides!

"expand $(x-(4+i))(x-(4-i))$"

So does it become: $x^2-8 x+17$?

4. Originally Posted by MathBlaster47
I Thank you heartily Pickslides!

"expand $(x-(4+i))(x-(4-i))$"

So does it become: $x^2-8 x+17$?
I like it.

5. Fantastic!
Thank you once more!

6. Originally Posted by pickslides

Quote:
Originally Posted by MathBlaster47

Q3:Use the factor theorem to determine whether (x-3) is a factor of
(I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

will be a factor if

now complete

is this zero?

A quick followup question:
If I want to do synthetic division on this function, would I be using 3 or -3?

7. Originally Posted by MathBlaster47
A quick followup question:
If I want to do synthetic division on this function, would I be using 3 or -3?
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

8. ## bad type setting on my part.

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?