Polynomial question. (1 Viewer)

[dasicmankev]

I'm stuck on question 28 Exercise 5E on the Patel 4 Unit book.

"Show that the polynomial x^n +mx - b = 0 has a multiple root provided (m/n)^n + (b/(n-1))^(n-1) = 0. Find this root."

I've tried solving it simultaneously but didn't arrive at the relation stated by the book.

Any help would be great.

 

[scardizzle]

since there is a multiple root then nx^n-1 + m = 0

therefore x^n-1 = -m/n - [1]

we also know that x(x^n-1 + m) = b

subbing in [1]:

x = b/(-m/n + m) = bn/m(n-1)

=) x^n-1 also = [bn/m(n-1)]^n-1

solving simultaneously with [1]:

[bn/m(n-1)]^n-1 = -m/n

therefore [b/(n-1)] ^ n-1 . [n/m] ^ n-1 = -m/n

therefore [b/(n-1)] ^ n-1 = -(m/n) ^n

therefore (m/n)^n + (b/(n-1))^(n-1) = 0

that's all i can do

 

[dasicmankev]

Thank you scardizzle. It hadn't occured to me to do it your way... nice work (Y).