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Grade: 12
if alpha and beta are the roots of x^3-x+1, show that alpha beta is a root of x^3+x^2-1 = 0
10 months ago

Answers : (3)

24739 Points
Dear student
Please check the question as it is wrong question as 1/alpha and 1/ beta will be the roots.
Please check and repost the question. You can also attach an imagery 
10 months ago
Vikas TU
11766 Points
Dear student 
The question is wrongly typed , please type the b part of quadratic eqution correctly . 
Please attach an image of the queston . 
We will help you . 
Good Luck 
10 months ago
Aditya Gupta
2043 Points
hello chandu, thanks for d ques. note that the question is perfectly fine. it is simply that vikas and arun couldnt ans it hence resorting to blaming the ques. the method of sol is a bit tricky, but i shall try my best to explain it.
let the roots of cubic x^3-x+1= 0 be a, b, c. obviously abc= – 1/1 or abc= p= – 1 (from theory of eqns).
now lets assume i want to form a new cubic eqn whose roots are ab, bc, ca, or equivalently abc/c, abc/a and abc/b or – 1/c, – 1/a, – 1/b. this means that if we replace x by – 1/x in the original cubic, its roots will be – 1/c, – 1/a, – 1/b. this eqn is ( – 1/x)^3 – ( – 1/x) + 1= 0
or 1/x^3= 1+1/x
or 1= x^3 + x^2
or x^3 + x^2 – 1= 0.
hence, the above eqn always has the roots alpha*beta, no matter what 2 – permutation among a, b, c is considered alpha and beta. 
kindly approve :)
10 months ago
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