f(x)=x3+2x2+x+1f(x) = x^3 + 2x^2 + x + 1f(x)=x3+2x2+x+1
Show that f(x)f(x)f(x) has a root between -1.5 and -2
Show that the equation f(x)=0f(x) = 0f(x)=0 can be written in the form x=−(x+1x2+2)\displaystyle x = -\left(\frac{x+1}{x^2} + 2\right)x=−(x2x+1+2)
Use the iteration formula xn+1=−(xn+1xn2+2)\displaystyle x_{n+1} = -\left(\frac{x_n+1}{x_n^2} + 2\right)xn+1=−(xn2xn+1+2) with x0=−2x_0 = -2x0=−2 to find, to 3 decimal places, the values of x1x_1x1, x2 x_2\,x2 and x3x_3x3.
By choosing a suitable interval, prove that α=−1.755\alpha = -1.755α=−1.755 to 3 decimal places
