Final answer:
To find a non-zero quadratic polynomial that is orthogonal to both p1(x) = 1 and p2(x) = x, calculate the inner products between p1, p2, and the unknown polynomial p. Solve the system of equations to find the values of a, b, and c that make the inner products zero. The correct answer is 2x² - 2/3.
Step-by-step explanation:
To find a non-zero quadratic polynomial that is orthogonal to both p1(x) = 1 and p2(x) = x under the L² inner product on the interval [-1,1], one can use the concept of orthogonality. Two vectors are orthogonal if their inner product is zero. In this case, we need to find a quadratic polynomial p(x) = ax^2 + bx + c, such that the inner product of p with p1 and p2 is zero.
Let's calculate the inner products:
Inner product of p1 and p:
∫(p1(x) * p(x)) dx
∫(1 * (ax^2 + bx + c)) dx
∫(ax^2 + bx + c) dx
(a/3)x^3 + (b/2)x^2 + cx
Inner product of p2 and p:
∫(p2(x) * p(x)) dx
∫(x * (ax^2 + bx + c)) dx
∫(ax^3 + bx^2 + cx) dx
(a/4)x^4 + (b/3)x^3 + (c/2)x^2
For both these inner products to be zero, we need to solve the following system of equations:
a/3 + b/2 + c = 0
a/4 + b/3 + c/2 = 0
Solving this system of equations will give us the values for a, b, and c that satisfy the given conditions. Evaluating the options given, the polynomial 2x² - 2/3 is the correct answer.