Set P₀(x) = 1 and P₁(x) = x. The Chebyshev polynomials (useful in approximation theory) are defined inductively by the formula P_n+1(x) = 2xP_n(x) − P_n−1(x).
(a) Show that P₂(x) = 2x² − 1.
(b) Compute P_n(x) for 3 ≤ n ≤ 6 using a computer algebra system or by hand, and plot y = P_n(x) over [−1, 1].
(c) Check that your plots confirm two interesting properties: (A) y = P_n(x) has n real roots in [−1, 1], and (B) for x ∈ [−1, 1], P_n(x) lies between −1 and 1.