Consider a function f given by a graph 2 y . 0 • bis [Select] 1 2 ·N Then, the second order Taylor polynomial approximating f(x) for x near 2 is given by P₂ (x) = a + b(x − 2) + c(x - 2)², where • a is [Select] and c is [Select] X

page 22 PAPER 1 5. FACTORS OF POLYNOMIALS J2013 1 It is given that f(x) = 6x³ - 5x² + ax + b has a factor of x + 2 and leaves a remainder of 27 when divided by x - 1. (i) Show that b = 40 and find the value of a. (ii) Show that f(x) = (x + 2)(px² +qx+r), where p, q and r are integers to be found. (iii) Hence solve f(x) = 0.

Q1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxX Given the data: Sheet III Chapter V: Interpolation & Data Fitting Interpolation & Data Fitting (Lagrange, Newton (divided, Forward) differences and Least squares X₁ 02 f(x) 0 Construct Lagrange interpolating polynomial, then find f(1.5), if you know the exact value of f(1.5) = -17.5, calculate the error ??? Q2 xxxxxxxxxxxxxxxxxxxXX Given the data: Q3 xxxxxxxxxxxxxx Given the data: 2 6 Xi f(x) Construct Newton's divided difference interpolating polynomial, then find f(1.5), if you know the exact value of f(1.5)= -17.5, calculate the error ??? -3 320 32 X₂ f(x) 48 96 -2 0 Xi -2 0 0 32 Construct Newton's forward difference interpolating polynomial, then find f(1.5). if you know the exact value of f(1.5)=-17.5, calculate the error ??? Q4 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Given the data: f(x)) Af(x)) 12 (a) 1+ 2x + x² (c) 1-2x + x² 0 Use an appropriate interpolating polynomial form to construct the interpolating…

Find the characteristic polynomial of A. Use x for the variable in your polynomial. You do not need to factor your polynomial. 3 -1 1 A=-9 3-3 -9 1 -1 characteristic polynomial of A is: 0