use newton method to find the root of x^4-5x^3+9x+3=0 accurate to six decimal places in the interval [4,6]

Question

use newton's method to find the root of x^4-5x^3+9x+3=0 accurate to six decimal places in the interval [4,6]

Answer 1

Differentiate the quartic: f'(x)=4x³-15x²+9 where f(x)=x⁴-5x³+9x+3

Starting with x₀=5 apply Newton’s method:

x₁=x₀-f(x₀)/f'(x₀)

x₁=5-f(5)/f'(5)=5-48/134=4.641791

x₂=4.537543...

x₃=4.528973...

x₄=4.528917...

x₅=4.52891795...

So to 6 dec places x=4.528918