Polynomials Question (1 Viewer)

[cutemouse]

Two of the roots of the equation x³-13kx²+13kx-1=0 are k and 1/k where k>0

(i) Find the 3rd root <--- I can do this, it's x=1

(ii) Find both values of k

The thing I don't get about (ii) is that I get k=1/3, but k=3 is also in the answers. Why is that?

 

[annabackwards]

It's because you just said that you'd let the roots be k and 1/k.

If you let k = 1/3, your 2 roots are 1/3 and 1/(1/3) = 3
If you let k = 3, your 2 roots are 3 and 1/3

So as you can see, k can be either 3 or 1/3

 

[Aerath]

Pretty much, since roots are k, 1/k, 1

Use sum of roots: k + 1/k + 1 = 13k
12k^2 - k - 1 = 0
(4k+1)(3k-1) = 0
k = 1/3 (k has to be positive)

If you let the first root be 1/3, then second root is 3, or vice versa (as annabackwards explained).