Determine whether the following set S is linear independent

Determine whether the following set S is linear independent or dependent.

{x^2 -1, 2x+5}

Show work.

Solution

You did not mention on which field you want to show L.i or l.d. , hence I am considering field of real R.

let c₁(x²-1)+c₂(2x+5)=0 implies that c₁x²+2c₂x+(5c₂-c₁)=0   where c₁,c₂ is in R .

now on the right hand side is zero polynomial so as in the left hand side(by the fundamental theorem of algebra) and hence all coffiecient is zero . that is c₁=c₂=0

so {x²-1 , 2x+5} is linearly independent over R