Question

Find the root of x³-x-1=0 , correct to four decimal place using Newton rapshn method

Answer 1

Find Initial Estimates for Roots

The Newton-Raphson method iterates and approaches one root. The cubic polynomial here has up to three roots to find.

f(x)=x3+x−1

f′(x)=3x2+1

The first derivative has x2 and the slope is always positive for this curve. The curve should only cross the x-axis once, i.e. one root.

x=(−1)⇒f(−1)=(−3)

x=0⇒f(0)=(−1)

x=+1⇒f(1)=1

The samples indicate the root occurs somewhere on the interval [0,1]. This means x=+1 is a good initial guess.