Estimate the zeros of the given function to the nearest hundredth

Question

h(x) = x^3 - 2x^2 - 14x + 30

^ = the exponent of the power for the number given

Answer 1

This can be solved using Newton's iteration method.

x_n+1=x_n-h(x_n)/h'(x_n) where the subscript denotes the iteration.

h'(x)=3x²-4x-14.

Let's start with x₀=0 and find out the nearest zero.

x₁=0-h(0)/h'(0)=0-30/-14=2.142857.

x₂=2.142857-h(2.142857)/h'(2.142857)=2.217434...

x₃=2.2205...., x₄=2.2205... So we have one solution to the nearest hundredth.

We can use synthetic division to reduce the cubic to a quadratic to find the other two solutions.

2.2205...| 1 -2            -14             |   30

                1 2.2205...  0.4896...   |  -30

                1 0.2205... -13.5103... |    0 = x²+0.2205x-13.5103.

Using the quadratic formula x=3.5670 and -3.7876. So the three zeroes to the nearest hundredth are:

2.22, 3.57, -3.79.