Question

Find a polynomial p3 such that {p0,p1,p2,p3} is an orthogonal basis.

A) p3(x)=x3
B) p3(x)=x3−x
C) p3(x)=x3+x2
D) p3(x)=x3−2x2+x

Answer 1

To find a polynomial p3 that is orthogonal to the basis polynomials p0, p1, and p2, we can use the Gram-Schmidt process. By subtracting the projections of p3 onto the previous basis polynomials, we can determine the polynomial that satisfies the condition.

Step-by-step explanation:

To find a polynomial p3 such that {p0,p1,p2,p3} is an orthogonal basis, we need a polynomial that is orthogonal to all the other basis polynomials (p0, p1, and p2). In this case, we can use the Gram-Schmidt process to find the orthogonal polynomial. Starting with p3(x) = x³, we subtract the projection of p3 onto the previous basis polynomials:

p3(x) = x³ - proj(p3, p0) - proj(p3, p1) - proj(p3, p2)

By substituting the given options for p3(x) and calculating the projections, we can determine which option satisfies the condition.

To find the projection, we use the inner product of two polynomials and divide by the inner product of the basis polynomial with itself:

proj(p3, p0) = (p3, p0) / (p0, p0)

proj(p3, p1) = (p3, p1) / (p1, p1)

proj(p3, p2) = (p3, p2) / (p2, p2)