What is the mathematic coding sequence for this problem AKA

What is the mathematic coding sequence for this problem? (AKA MATLAB PROGRAM)

Determine the quartic f (x) = ax^(4) + bx^(3) + cx^(2) + dx + e that passes through the five points (-2:5;-62), (-1:5;-7:2), (-0:5;8:3), (1;3:7) , (3;45:7) by solving a linear system. Corroborate your result by graphing the quartic and the points on the same plot.

Solution

Given a general fourth degree equation x⁴ + bx³ + cx² + dx + e = 0, you can rearrange terms to form the equation

x⁴ + bx³ = -cx² - dx - e.

Now add the expression (b²/4 + 2p)x² + bpx + p² to both sides:

x⁴ + bx³ + (b²/4 + 2p)x² + bpx + p² = (b²/4 + 2p - c)x² + (bp - d)x + p² - e.

The left hand side is now a perfect square: (x² + (b/2)x + p)². You want to find a real number p such that the right hand side is also a square. In order for the right hand side to be a square quadratic, the discriminant must be zero. That is,

(bp - d)² - 4(b²/4 + 2p - c)(p² - e) = 0.

This simplifies to a cubic equation in p:

-8p³ + 4cp² + (8e - 2bd)p + d² - 4ce + b²e = 0.

Since every cubic equation has at least one real root, you can find a suitable value of p to resolve the quartic. After you plug in the value of p, you take the square root of both sides to create two quadratic equations. This gives you a total of four solutions

Example: Solve the quartic equation x⁴ - 4x³ + 5x² - 4x + 4 = 0.

x⁴ - 4x³ = -5x² + 4x - 4
x⁴ - 4x³ + (4 + 2p)x² - 4px + p² = (4 + 2p)x² - 4px + p² -5x² + 4x - 4

Now solve the cubic -8p³ + 20p² = 0. The solutions are p = 0, 0, 5/2. You can use any real value of p to plug into the quartic. For this example, we will use 0 since it is easier to work with.

x⁴ - 4x³ + 4x² = -x² + 4x - 4
sqrt(x⁴ - 4x³ + 4x²) = sqrt(-x² + 4x - 4)
x² - 2x = ±i(x - 2)

This yields two quadratic equations with complex coefficients:

x² + (-2 + i)x - 2i = 0
x² + (-2 - i)x + 2i = 0

Using the quadratic equation, the roots of the first equation are 2 and -i, and the roots of the second are 2 and i. These four roots are the roots of the original quartic.