+0  
 
0
36
3
avatar+1961 

What is (2x^5 - 3x^4 + 5x^3 - 10x^2 - 12x + 8) divided by (x^2 + 4)? I’m in pre-calc (in case that helps) and the original problem says

 

Solve over the complex numbers. 

 2x^5 - 3x^4 + 5x^3 - 10x^2 - 12x + 8 = 0  if 2i is a root 

 

So what I did was multiply (x - 2i)(x + 2i) since they are factors (if there’s positive 2i, there has to be a negative one to cancel out the i), and I was going to factor out x^2 + 4, but I’m having trouble. Thank you in advance! :)

 
saseflower  Nov 14, 2017
Sort: 

3+0 Answers

 #1
avatar+78753 
+3

(2x^5 - 3x^4 + 5x^3 - 10x^2 - 12x + 8) divided by (x^2 + 4)?

 

 

                   2x^3   - 3x^2   - 3x    + 2

x^2 + 4  [    2x^5    - 3x^4  + 5x^3  - 10x^2    - 12x   +  8 ]

                   2x^5                + 8x^3

                  ____________________________________ 

                              -3x^4   -  3x^3  - 10x^2

                              -3x^4                - 12x^2

                  ____________________________________

                                            -3x^3  +  2x^2   - 12x

                                            -3x^3                  -12x

                  _____________________________________

                                                          2x^2               +  8

                                                          2x^2               +  8

                   ______________________________________

                                                                     ( No remainder )

 

 

 

 

 2x^5 - 3x^4 + 5x^3 - 10x^2 - 12x + 8 = 0

 

If   2i is a root...so is  -2i

 

So  we have this  (x - 2i ) ( x + 2i)  =  x^2 - 4i^2  =   x^2 + 4 is a factor

 

We already divided this...let's use what we found to find the other three roots

 

 2x^3   - 3x^2   - 3x    + 2  = 0   

 

 Note that we can use a factoring trick here, saseflower

 

Write   -3x^2  as -4x^2 +  x

 

2x^3 - 4x^2  + x - 3x + 2  =  0  factor

 

2x^2 ( x - 2)  + ( x - 2) ( x - 1)  = 0

 

(x -2)  [ 2x^2  + x - 1 ]  = 0

 

(x - 2) (2x - 1) ( x + 1)  = 0

 

So...setting these factors to 0 and solving for x gives us the other three roots of

 

x = 2 , x = 1/2   and x = -1

 

 

cool cool cool

 

 

 

 

 

 

 

 

 

 

  

 
CPhill  Nov 14, 2017
 #3
avatar+91049 
+1

You have the setting out of these off pat Chis :)  I am impressed!

As you know I have given up and just do them on paper and take a photo. 

 
Melody  Nov 14, 2017
 #2
avatar+1961 
+2

Thank you so much! :)

 
saseflower  Nov 14, 2017

10 Online Users

avatar
avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details