Proving Linear Independence of Polynomials with Non-Zero Degrees | Math Solution

[kash25]

Linear independence!?

Homework Statement

Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1.

The Attempt at a Solution

I am pretty sure the statement to prove is incorrect.
If we use deg(p) = -1 and deg(q) = -2, we can easily show that the two are linearly independent (consider the functions p(x) = 1/x and q(x) = 1/x^2).
We can have k/x + l/x^2 = 0
then kx + l = 0.
Then we can differentiate and get k = 0 and l = 0, which disproves the statement.
How does this make any sense?

 

[Dick]

If you are talking about polynomials you are only talking about linear combinations of x^n where n>=0. There is so much more wrong with your counterexample, I don't know where to start... The statement is true, now try and prove it.