Determine a root of the polynomial function using Muller’s Method...

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This is likely to be a lengthy solution so I may need to abbreviate some of the details. But first we need some preliminaries. Here is a table and a graph. f(x) is shown in red. The blue curve is a parabola attempting to match the shape of the curve using three points near to a zero. The roots, or zeroes, are the 4 points of intersection of the red curve and the x-axis.

 

 

 

 

Now to find the parabola’s equation. There are various ways to do this but I’ll use simultaneous equations.

g₁(x)=ax²+bx+c. To find a, b, c using x₀=-15, x₁=-10, x₂=-5.

(1) 40.2926=225a-15b+c

(2) -35.6704=100a-10b+c

(3) -12.2184=25a-5b+c

(1)-(2): 75.963=125a-5b-151.926=-250a+10b

(1)-(3): 52.511=200a-10b

Adding the last two equations: -99.415=-50a, a=1.9883.

From this b=34.5149, c=110.6486.

g(x)=1.9883x²+34.5149x+110.6486 (blue curve).

More to follow...

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