polynomials P21 = x^2 + 2x – 1, P22 =2x^2-3x, p23= x-1 are independent

Question

To show that the system of polynomials P21 = x^2 + 2x – 1, P22 =2x^2-3x, p23= x-1 are independent

Answer 1

There are no common zeroes within the system:

P₂₁ has no rational factors; P₂₂ factorises x(2x-3), so its two zeroes are 0 and 3/2; x-1 is linear with zero 1.

There is no common intersection, although P₂₁ and P₂₂, P₂₁ and P₂₃, and P₂₂ and P₂₃ intersect:

P₂₁ and P₂₂: x²+2x-1=2x²-3x, x²-5x+1=0 has two intersection points (at (5+√21)/2 and (5-√21)/2);

P₂₁ and P₂₃: x²+2x-1=x-1, x²+x=0 has two intersection points (at 0 and -1);

P₂₂ and P₂₃: 2x²-3x=x-1, 2x²-4x+1=0 has two intersection points (at 1+√2/2 and 1-√2/2).

Conclusion: the polynomials are independent.