+0  
 
0
68
1
avatar+525 

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
Sort: 

1+0 Answers

 #1
avatar+82546 
+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

17 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