Question

Consider the following points.

(−1, 3), (0, 0), (1, 1), (4, 52)

(a) Determine the polynomial function of least degree whose graph passes through the given points.

Consider the following points.

(−2, 44), (−1, 0), (0, −14), (1, −16), (2, 0)

(a) Determine the polynomial function of least degree whose graph passes through the given points.

Answer 1

To find the polynomial function of least degree that passes through the given points, we need to use the method of interpolation. We can use the Lagrange interpolation formula to determine the polynomial.

Step-by-step explanation:

To find the polynomial function of least degree that passes through the given points, we need to use the method of interpolation. We can use the Lagrange interpolation formula to determine the polynomial. The formula is given by:

P(x) = (x - x2)(x - x3)(x - x4) + (x - x1)(x - x3)(x - x4)y2 + (x - x1)(x - x2)(x - x4)y3 + (x - x1)(x - x2)(x - x3)y4

where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the given points. Plugging in the given points (-1, 3), (0, 0), (1, 1), (4, 52), we can find the polynomial function.

For the second set of points, we need to use the same method of interpolation. We can again use the Lagrange interpolation formula to determine the polynomial. The formula will be similar to the first one but with five terms instead of four. Plugging in the given points (-2, 44), (-1, 0), (0, -14), (1, -16), (2, 0), we can find the polynomial function.

Answer 2

(a)
`P(x)=0.4x^(3) +2x^(2) -1.4x`

(b)
`P(x)=x^(4) -x^(3) +5x^(2) -7x-14`

Step-by-step explanation:

Let´s use Divided Differences Method of Polynomial Interpolation given by this iteration:

`f[x_k,x_k_+_1,...,x_k_+_i]=(f[x_k_+_1,x_k_+_2,...,x_k_+_i]-f[x_k,x_k_+_1,...,x_k_+i_-_1])/(x_k_+_i-x_k)`

k∈[0,n-i]

Thus the Newton polynomial can be written as:

`P_n_-_1(x)=f[x_0]+f[x_0,x_1](x-x_0)+f[x_o,x_1,x_2](x-x_0)(x-x_1)+...+f[x_n,x_n_-_1,...,x_1](x-x_n)(x-x_n_-_1)...(x-x_1)`

(a) I attached you the procedure in the first table, using it we have:

`P(x)=3+(-3)(x+1)+(2)(x+1)(x)+0.4(x+1)(x)(x-1)`

Operate P(x) using the distributive property:

`P(x)=0.4x^(3) +2x^(2) -1.4x`

(b) I attached you the procedure in the second table, using it we have:

`P(x)=44+(-44)(x+2)+(15)(x+2)(x+1)+(-3)(x+2)(x+1)(x)+(1)(x+2)(x+1)(x)(x-1)`

Operate P(x) using the distributive property:

`P(x)=x^(4) -x^(3) +5x^(2) -7x-14`