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 states that for a set of n + 1 distinct points (x0, y0), (x1, y1), ..., (xn, yn), there exists a unique polynomial of degree at most n that passes through these points.
In this case, we have four points, which means we are looking for a polynomial of degree 3. Let's denote the polynomial as P(x).
The formula to find P(x) using Lagrange's interpolation formula is:
P(x) = Σ(yi * Li(x))
Where Σ denotes the summation from i = 0 to n, yi represents the y-coordinate of the ith point, and Li(x) is the Lagrange polynomial defined as:
Li(x) = Π(x - xj) / Π(xi - xj) for j ≠ i
Where Π denotes the product from j = 0 to n, xi represents the x-coordinate of the ith point, and xj represents the x-coordinate of the jth point.
Using this formula, we can calculate P(x) step by step:
P(x) = (-12 * L0(x)) + (0 * L1(x)) + (6 * L2(x)) + (12 * L3(x))
where L0(x), L1(x), L2(x), and L3(x) are the Lagrange polynomials for each point (0,-12), (1,0), (3,6), and (4,12) respectively.
L0(x) = ((x - 1)(x - 3)(x - 4)) / ((0 - 1)(0 - 3)(0 - 4))
L1(x) = ((x - 0)(x - 3)(x - 4)) / ((1 - 0)(1 - 3)(1 - 4))
L2(x) = ((x - 0)(x - 1)(x - 4)) / ((3 - 0)(3 - 1)(3 - 4))
L3(x) = ((x - 0)(x - 1)(x - 3)) / ((4 - 0)(4 - 1)(4 - 3))
Simplifying each Lagrange polynomial:
L0(x) = (x^3 - 8x^2 + 19x - 12) / (-12)
L1(x) = -(2x^3 - 11x^2 + 18x) / 3
L2(x) = (3x^3 - 15x^2 + 16x) / 6
L3(x) = -(2x^3 - 6x^2 + 3x) / 24
Substituting the Lagrange polynomials into the polynomial equation:
P(x) = (-12 * ((x^3 - 8x^2 + 19x - 12) / (-12))) + (0 * (-(2x^3 - 11x^2 + 18x) / 3)) + (6 * ((3x^3 - 15x^2 + 16x) / 6)) + (12 * (-(2x^3 - 6x^2 + 3x) / 24))
Simplifying further:
P(x) = (x^3 - 8x^2 + 19x - 12) - (2x^3 - 11x^2 + 18x) + (3x^3 - 15x^2 + 16x) - (2x^3 - 6x^2 + 3x)
Combining like terms:
P(x) = x^3 - 8x^2 + 19x - 12 - 2x^3 + 11x^2 - 18x + 3x^3 - 15x^2 + 16x - 2x^3 + 6x^2 - 3x
Simplifying further:
P(x) = 2x^3 - 6x^2 + 19x - 12
Therefore, the third-degree polynomial that passes through the given points is P(x) = 2x^3 - 6x^2 + 19x - 12.