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(a) Given  \(3x+5y-6z=2\)\(5xy-10yz-6xz=-41\) , and\(xyz=6\)
Let a=3x, b=5y, and x=-6z
Determine the monic cubic polynomial in terms of a variable  whose roots are t=a, t=b, and t=c. Make sure to enter your answer in terms of t and only t, in expanded form.

 

 

(b) Given that (x, y, z) is a solution to the original system of equations, determine all distinct possible values of x+y.
 

CrystalRead  Nov 12, 2018
 #1
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I think you mean c = -6z. Assuming this is so, then:

 

(t-a)(t-b)(t-c) = 0

 

t3 - (a + b + c)t2 + (ab + bc + ac)t - abc = 0

 

a + b + c = 2

 

(ab + bc + ac)/3 = -41

 

abc/(3*5*(-6)) = 6

 

t3 - 2t2 - 123t + 540 = 0

 

See if you can now do part b.

Alan  Nov 12, 2018

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