Let V be the vector space over R consisting of all polynomials in the variable x withreal coefficients. Determine whether the polynomialsf (x) = x2 + 3, g(x) = −x2 + x + 2, h(x) = x2 − 2x − 7are linearly independent

To determine if the polynomials f(x) = x^2 + 3, g(x) = -x^2 + x + 2, and h(x) = x^2 - 2x - 7 are linearly independent, we need to set up and solve the following equation:

af(x) + bg(x) + c*h(x) = 0

This gives us:

a*(x^2 + 3) + b*(-x^2 + x + 2) + c*(x^2 - 2x - 7) = 0

Simplifying, we get:

(a - b + c)x^2 + bx - 2cx + 3a + 2b - 7c = 0

For this equation to hold for all x, the coefficients of the x^2, x, and constant terms must all be zero. This gives us the following system of equations:

a - b + c = 0 b - 2c = 0 3a + 2b - 7c = 0

Solving this system of equations will give us the values of a, b, and c. If a, b, and c are all zero, then the polynomials are linearly independent. If any of a, b, or c are non-zero, then the polynomials are linearly dependent.

Solving the system of equations, we find that a = 0, b = 0, and c = 0. Therefore, the polynomials f(x), g(x), and h(x) are linearly independent.