Polynomial Question need Help (1 Viewer)

[kevinsta]

If alpha and beta are the roots of x^2 - 5x -5 = 0 find a^2+b^2, a^3+b^3

Is there a way i can manipulate the question to solve it or is there a rule that i must follow.

 

[braintic]

If alpha and beta are the roots of x^2 - 5x -5 = 0 find a^2+b^2, a^3+b^3

Is there a way i can manipulate the question to solve it or is there a rule that i must follow.

Since alpha and beta are roots, then:
α^2 - 5α - 5 = 0
β^2 - 5β - 5 = 0

Adding:
(a^2 + b^2) - 5(a+b) - 10 = 0

Find a+b from sum of roots, then substitute to get a^2 + b^2.


Since alpha and beta are roots of x^2 - 5x -5 = 0, then they are also roots of x^3 - 5x^2 - 5x = 0
Then repeat the process, using your previous answer.

 

[iStudent]

Alternatively,
a^2+b^2 = (a+b)^2-2ab
Use sum/product of roots and evaluate
also
(a+b)^3 = a^3+b^3+3ba^2+3ab^2
so a^3+b^3 = (a+b)^3 - 3ab(a+b)
Use sum/product and evaluate

 

[braintic]

Alternatively,
a^2+b^2 = (a+b)^2-2ab
Use sum/product of roots and evaluate
also
(a+b)^3 = a^3+b^3+3ba^2+3ab^2
so a^3+b^3 = (a+b)^3 - 3ab(a+b)
Use sum/product and evaluate

He should learn the other method ... there are induction questions that require it.