The quadrature rule [“ f(x)dx = w₁ f(x1) + w2f(x2) + E (2) -h can be made exact for all cubic polynomials, by using a suitable choice of weights and points, respectively wi and xi, i = {1, 2}. • Write down four equations satisfied by w₁, W2, x1 and x2. • Combine two of these equations to show that x2 = -x₁ and hence that w₁ = W2. • Calculate the weights and then the value of the quadrature points. ⚫ Find the error term and write the quadrature rule (2). What class of quadrature rules does it belong to?

A quadratic polynomial y = and = 4x² +7x 3 has roots p and q. A new quadratic with roots is given by g(x) = x² + ax + ß. r = q Express a and ß as fractions. α= P B =

For the polynomial below, 1 is a zero. h (x)=x' 7x + 10x 4 | Express h (x) as a product of linear factors. h (x) = 0

A quadratic polynomial y = 4x² + 7x – 3 has roots p and q. Find a quadratic g(x) = x2 + ax + ß with roots r = ! and s = !. a. g(x) 3D x? + 쫓-을 O b. g(x) = x² + 4 + g(x) %=D x? - 좋 + d. g(x) = x² – - C.