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

a.
`f(x) = 0.4x^3+2x^2-1.4x\\\\`

b.
`f(x) = x^4-x^3+5x^2-7x-14\\\\`

Explanation:

A polynomial of degree three satisfies the four points offered, in this way, if we evaluate said polynomial in the points, we can generate a linear system of four equations with four incognites that when solved will allow us to know the coefficients.

`f(x) = ax^3+bx^2+cx+d\\\\`

`f(-1) = a(-1)^3+b(-1)^2+c(-1)+d = -a+b-c+d = 3\\\\f(0) = a(0)^3+b(0)^2+c(0)+d = d = 0\\\\f(1) = a(1)^3+b(1)^2+c(1)+d = a+b+c+d = 1\\\\f(4) = a(4)^3+b(4)^2+c(4)+d = 64a+16b+4c+d = 52`

The linear system is:

`-a+b-c+d = 3\\\\d = 0\\\\a+b+c+d = 1\\\\64a+16b+4c+d = 52\\\\`

The solution of the linear system is:

`a=0.4\\\\b=2\\\\c=-1.4\\\\d=0\\\\`

Then the function is:

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

b. A polynomial of degree four satisfies the five points offered, in this way, if we evaluate said polynomial in the points, we can generate a linear system of five equations with five incognites that when solved will allow us to know the coefficients.

`f(x) = ax^4+bx^3+cx^2+dx+e\\\\`

`f(-2) = a(-2)^4+b(-2)^3+c(-2)^2+d^(-2)+e = 16a-8b+4c-2d+e= 44\\\\f(-1) = a(-1)^4+b(-1)^3+c(-1)^2+d^(-1)+e = a-b+c-d+1 = 0\\\\f(0) = a(0)^4+b(0)^3+c(0)^2+d(0)+e = e = -4\\\\f(1) = a(1)^4+b(1)^3+c(1)^2+d(1)+e = a+b+c+d+e = -16\\\\f(2) = a(2)^4+b(2)^3+c(2)+d(2)+e = 16a+8b+4c+2d+e = 0`

The linear system is:

`16a-8b+4c-2d+e= 44\\\\a-b+c-d+1 = 0\\\\e = -4\\\\a+b+c+d+e = -16\\\\16a+8b+4c+2d+e = 0`

The solution of the linear system is:

`a=1\\\\b=-1\\\\c=5\\\\d=-7\\\\e=-14`

Then the function is:

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