A third-degree polynomial passes through the points (0,-12), (1,0) ,(3,6) and (4,12).
a). Determine this polynomial using Lagrange's interpolation formula
Lagrange's interpolation formula can be used to find a polynomial of degree n that passes through n+1 given points. In this case, we have 4 points, so we will be finding a third-degree polynomial.
We can use the following formula to calculate the polynomial:
P(x) = (x - x1)(x - x2)(x - x3)(x - x4)[y1/l1 + y2/l2 + y3/l3 + y4/l4]
where xi and yi are the coordinates of the given points, and li is the Lagrange coefficient given by:
li = (x - x1)(x - x2)(x - x3)(x - x4)/(xi - x1)(xi - x2)(xi - x3)(xi - x4)
Using the given points, we have:
x1 = 0, y1 = -12
x2 = 1, y2 = 0
x3 = 3, y3 = 6
x4 = 4, y4 = 12
Now, let's calculate the polynomial using the formula:
P(x) = (x - 0)(x - 1)(x - 3)(x - 4)[-12/(0 - 0)(0 - 1)(0 - 3)(0 - 4) + 0/(1 - 0)(1 - 1)(1 - 3)(1 - 4) + 6/(3 - 0)(3 - 1)(3 - 3)(3 - 4) + 12/(4 - 0)(4 - 1)(4 - 3)(4 - 4)]
Simplifying the formula:
P(x) = x(x - 1)(x - 3)(x - 4)[-12/(0)(-1)(-3)(-4) + 0/(1)(1-1)(1-3)(1-4) + 6/(3)(3-1)(3-3)(3-4) + 12/(4)(4-1)(4-3)(4-4)]
P(x) = x(x - 1)(x - 3)(x - 4)[6/(-6) + 6/(-6) + 6/12]
P(x) = x(x - 1)(x - 3)(x - 4)[-1 + 1 + 0.5]
P(x) = x(x - 1)(x - 3)(x - 4)(0.5)
Expanding and simplifying:
P(x) = 0.5x^4 - 4x^3 + 9.5x^2 - 6x
Therefore, the polynomial that passes through the given points is:
P(x) = 0.5x^4 - 4x^3 + 9.5x^2 - 6x