Answer: To find the zeros of the polynomial p(x) = 3x^3 - 10x^2 + 10x - 4, we can use various methods such as factoring, synthetic division, or numerical methods. In this case, let's use a numerical method called the Newton-Raphson method to approximate the zeros.
We start by selecting an initial guess for the root. Let's start with x = 1.
Apply the Newton-Raphson formula to refine the estimate of the root:
x1 = x0 - (p(x0) / p'(x0)),
where x1 is the refined estimate, x0 is the initial guess, p(x0) is the value of the polynomial at x0, and p'(x0) is the derivative of the polynomial at x0.
Repeat step 2 until the estimate converges to the actual root.
Applying these steps, we can find the zeros of p(x) as follows:
Initial guess: x = 1
Using the Newton-Raphson method:
x1 = x0 - (p(x0) / p'(x0))
For p(x) = 3x^3 - 10x^2 + 10x - 4:
p'(x) = 9x^2 - 20x + 10
Using the initial guess x = 1:
x1 = 1 - (3(1)^3 - 10(1)^2 + 10(1) - 4) / (9(1)^2 - 20(1) + 10)
= 1 - (3 - 10 + 10 - 4) / (9 - 20 + 10)
= 1 - (-1) / (-1)
= 1 + 1
= 2
So, the refined estimate of the root is x = 2.
Next, we repeat the Newton-Raphson method with the refined estimate x = 2:
x2 = 2 - (3(2)^3 - 10(2)^2 + 10(2) - 4) / (9(2)^2 - 20(2) + 10)
Calculating x2:
x2 = 2 - (24 - 40 + 20 - 4) / (36 - 40 + 10)
= 2 - (0) / (6)
= 2
The estimate has converged, and we find that x = 2 is a zero of the polynomial p(x).
So, the zero of p(x) = 3x^3 - 10x^2 + 10x - 4 is x = 2, with a multiplicity of 1.
Therefore, the zeros of p(x) are x = 2.