Algebra> if alpha and beta are the roots of x^3-x+...

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

Chandu , 5 Years ago

Grade 12

3 Answers

Arun

Last Activity: 5 Years ago

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

Vikas TU

Last Activity: 5 Years ago

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

Cheers

Aditya Gupta

Last Activity: 5 Years ago

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 :)