Three tricky questions that need solving..

• Jan 5th 2010, 01:08 PM
MathBlaster47
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).

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.)

• Jan 5th 2010, 01:27 PM
pickslides
Hi there MathBlaster47

Quote:

Originally Posted by MathBlaster47

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...)

Using the complex conjugate, expand the following

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

Quote:

Originally Posted by MathBlaster47

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

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

Quote:

Originally Posted by MathBlaster47

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.)

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

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

is this zero?
• Jan 5th 2010, 01:45 PM
MathBlaster47
Quote:

Originally Posted by pickslides
Hi there MathBlaster47

Using the complex conjugate, expand the following

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

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

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

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

is this zero?

I Thank you heartily Pickslides!

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

So does it become: $\displaystyle x^2-8 x+17$?
• Jan 5th 2010, 02:25 PM
pickslides
Quote:

Originally Posted by MathBlaster47
I Thank you heartily Pickslides!

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

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

I like it.
• Jan 5th 2010, 02:39 PM
MathBlaster47
Fantastic!
Thank you once more! (Rock)
• Jan 6th 2010, 01:49 PM
MathBlaster47
Quote:

Originally Posted by pickslides

Quote:
Originally Posted by MathBlaster47 http://www.mathhelpforum.com/math-he...s/viewpost.gif

Q3:Use the factor theorem to determine whether (x-3) is a factor of http://www.mathhelpforum.com/math-he...e7412eb4-1.gif
(I don't understand understand the Factor Theorem very well, so I find myself thoroughly lost.)

will be a factor if http://www.mathhelpforum.com/math-he...9abcd813-1.gif

now complete http://www.mathhelpforum.com/math-he...8a71b345-1.gif

is this zero?

A quick followup question:
If I want to do synthetic division on this function, would I be using 3 or -3?
• Jan 6th 2010, 01:55 PM
e^(i*pi)
Quote:

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
• Jan 6th 2010, 02:36 PM
MathBlaster47
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?