F(x)=3x^4-x^3-9x^2+159x-52 Solution

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To find the solutions of the polynomial function f(x)=3x4−x3−9x2+159x−52, we can use various methods such as factoring, synthetic division, or numerical methods like the Rational Root Theorem or graphing.

In this case, let’s utilize numerical methods to approximate the roots of the polynomial.

One common numerical method is Newton’s method (also known as Newton-Raphson method), which provides iterative approximation to the roots of a real-valued function.

Newton’s method involves iterating the formula:

xn+1​=xn​−f′(xn​)f(xn​)​

Where:

  • xn+1​ is the next approximation of the root.
  • xn​ is the current approximation of the root.
  • f(xn​) is the value of the function at xn​.
  • f′(xn​) is the derivative of the function at xn​.

We start with an initial guess x0​ and iteratively apply the formula until convergence to the desired accuracy.

Let’s proceed with Newton’s method to approximate the roots of f(x):

  1. We first need to find the derivative of f(x), denoted as f′(x), which is necessary for Newton’s method.
  2. f′(x)=12x3−3x2−18x+159
  3. Next, we choose an initial guess for the root, denoted as x0​. This can be any reasonable value close to the actual root.
  4. Then, we apply Newton’s method iteratively using the formula mentioned above until convergence.

Let’s perform the iterations:

For the first derivative, f′(x)=12x3−3x2−18x+159, and for simplicity, we’ll choose an initial guess x0​=1.

Iteration 1:

x1​=1−12(1)3−3(1)2−18(1)+1593(1)4−(1)3−9(1)2+159(1)−52​

x1​≈1.41176470588

Iteration 2:

x2​=x1​−12(x1​)3−3(x1​)2−18(x1​)+1593(x1​)4−(x1​)3−9(x1​)2+159(x1​)−52​

x2​≈1.41309981354

By continuing this process, we can approximate the root of the polynomial function f(x). Repeat the iterations until the desired level of accuracy is achieved.

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