MATH 2050 Linear Algebra

(Section 4)
2009 Winter

Assignment 1, Question 5   -   Alternate Solution

  1. Find the conditions on a, b, c, d, so that the cubic curve   y = ax3 + bx2 + cx + d   passes through all four points (–1, –8), (0, 1), (1, 2) and (2, 13).

    The equation of the cubic curve is satisfied by all four points:
    -a + b - c + d = -8 0 + 0 + 0 + d = 1 a + b + c + d = 2 8a + 4b + 2c + d = 13
    Equation 2 tells us immediately that   d = 1.
    Substituting this value into the other three equations and rearranging them, we find the equivalent system
    a + b + c = 1 -a + b - c = -9 8a + 4b + 2c = 12
    [row reduction]
    [row reduction]
    [row reduction]
    and we again obtain the unique solution   a = 3 , b = –4 , c = 2 and d = 1 .

      Return to the Solutions to Assignment 1   [To the Solutions to Assignment 1.]
      Back to previous page   [Return to your previous page]

      Created 2008 12 31 and most recently modified 2009 01 02 by Dr. G.H. George