The number of basis vectors needed for a quadratic polynomial is equal to the degree of the polynomial plus one. In this case, since the polynomial is of degree 4, we will need 5 basis vectors.
To find basis vectors for a quadratic polynomial, we can use the standard basis vectors 1, x, x^2, x^3, and x^4. Then, we can use Gaussian elimination to reduce the matrix formed by these vectors to row echelon form. The remaining vectors will be our basis vectors.
Yes, you can use any set of linearly independent vectors as basis vectors for a quadratic polynomial. However, the standard basis vectors are the most commonly used and easiest to work with.
Finding basis vectors for a quadratic polynomial allows us to express any other quadratic polynomial as a linear combination of these basis vectors. This can make solving equations and performing other operations much easier.
No, there is no specific order in which the basis vectors should be arranged. However, it is common to arrange them in ascending powers of x, with the constant term as the first basis vector.