+0  
 
0
307
1
avatar+644 

In this multi-part problem, we will consider this system of simultaneous equations:
3x+5y-6z =2, 
5xy-10yz-6xz = -41, 
xyz = 6. 

Let a=3x, b=5y and c=-6z .

Determine the monic cubic polynomial in terms of a variable t whose roots are ,t=a, t=b, and t=c.

waffles  Jan 18, 2018
 #1
avatar+89715 
+1

x  = a/3      y   =  b/5       z   =  - c/6

 

a  + b  +  c   =  2

5 (a/3)(b/5)  -  10(b/5)(-c/6)  - 6(a/3)(-c/6)  = -41

abc / -90 =  6

 

a +  b  +  c  =  2       ⇒   a + b  =  2 - c       (1)   

ab  + bc  +  ac  =  -123    ⇒    ab  + c ( a + b)  =  -123    (2) 

abc   =  -540   ⇒   ab  =  -540/c        (3)

 

Sub  (1)  and (3)  into (2)

 

-540/ c  + c (2 - c)  =  -123

-540/c   + 2c - c^2  = -123

-540 + 2c^2 - c^3  = -123c

c^3 - 2c^2  - 123c + 540  = 0     re-write as

c^3 -  2c^2  - 63c  -  60c + 540  = 0

c(c^2 - 2c - 63)   -  60c + 540  = 0

c(c -9)(c + 7) -  60 (c - 9) =  0

(c - 9) [ c(c + 7) - 60 ]  = 0

(c - 9) [ c^2 + 7c - 60] =  0

(c - 9) ( c + 12)(c - 5)   =  0

 

So...possible values of c =   5, 9  and - 12

z  is an integer if  c = -12

And  ab  =  45

And  a + b  =  c  - (- 12)  ⇒  a +  b  =  14

And x is an ineger if a  = 9

And  y is an integer if b  =  5

 

So the cubic monomial we need is

 

(t + 12) (t -9) (t - 5)   =  

 

t^3 - 2 t^2 - 123 t + 540

 

 

cool cool cool

CPhill  Jan 18, 2018

30 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.