In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2) = v, P(3) = w. 1) Let P₁(x) = (-2)(1-3). P₂(x) = (x-1)(x-2) (3-1)(3-2). Prove that the polynomial P(x) = uP₁(x) +vP₂(x) + wP3(x) is a solution to our problem. (x-1)(x-3) 2-3 (2-1)(2-3) and P3(x) = =

Show that if ax2+bx+c=0 for all x, then a=b=c=0.

In this problem we prove, using linear algebra, that for any u, v, w there is a unique polynomial P(x) of degree at most 2 such that P(1) = u, P(2) = v, P(3) = w. Let P₁(x) = P₂(x) = (x-2)(x-3) (1-2)(1-3)* (x-1)(x-3) (2-1)(2-3) and P3(x) 2) Prove that the list P₁, P2, P3 is a basis of P₂ (R). - (x-1)(x-2) (3-1)(3-2)*

This is a question from a linear algebra course: Let V = R[X]3, the polynomials of degree at most three, and B = {1, X, X2, X3}. Show what the image under fB is of:• the four basic elements: P1(X) = 1, P2(X) = X, P3(X) = X2 and P4(X) = X3• P(X) = 2 + 6X + 3X2 + 4X3

Prove that you can find a polynomial p(x) = Ax2 + Bx + C that passes through any three points (x1, y1), (x2, y2), and (x3, y3), where the xi ’s are distinct.

2.)

In P₂, let (p(x), g(x)) = p(−1)q(−1) +p(0)q(0) +p(1)q(1). Let U = span{x², x + 1}. Find the polynomial in U that is closest to f(x) = 4x + 4. 1²+ x+

Let P0" (x) denote the quadratic polynomial that interpolates the data {(xo, yo), (x1, yı), (x2, y2)}; let P"."(x) denote the quadratic polynomial that interpolates the data {(x1, yı), (x2, y2), (x3, Y3)}. Finally, let P3(x) de- note the cubic polynomial interpolating the data{(xo, yo), (x1, yı), (x2, y2), (x3, y3)}. Show that 14. (a) (x3 – x)P0.2) (x) + (x – xo) P!.3) (x) P3 (x) = (1,3) X3 - Xo How might this be generalized to constructing P(x), interpolating {(xo, yo), (Xn, Yn)}, from interpolation polynomials of degreen – 1? (b) ....

Using Newton's Divided Difference, the polynomial that interpolates x + sinx at the points x, = 1,x2 = 2 ,x3 = 3,x4 = 4 , and xs = 5 can be written as P(x) = c, + c,(x - 1) + c,(x – 1)(x – 2) + cz (x – 1)(x – 2)(x – 3) +C4 (х — 1)(х — 2)(х — 3)(х — 4). What is c, ? A 1.84147 B 0.004970 -0.649329 3.36812

, 20-20. Find the polynomial function f (x) given that f(0) = 2, f(1)= 3, f (2) = 12 and f (3) = 35. Hence find f(5).