estimate, correct to three decimal places all the zeros of the polynomial p(X) = x^4 - 4x^3 +5x -1
To estimate the zeros of a polynomial, we can use numerical methods such as the Newton-Raphson method or graphical methods like drawing the graph of the polynomial. Let's use the Newton-Raphson method to estimate the zeros of the polynomial p(x) = x^4 - 4x^3 + 5x - 1.
The Newton-Raphson method requires an initial guess for each zero. We can start with an initial guess of x = 1 for the first zero.
Step 1: Calculate the derivative of the polynomial p(x). The derivative of p(x) = x^4 - 4x^3 + 5x - 1 is p'(x) = 4x^3 - 12x^2 + 5.
Step 2: Using the initial guess x=1, apply the Newton-Raphson formula:
x_new = x_old - f(x_old)/f'(x_old)
For the first zero, let's use the initial guess x_old = 1.
x_new = x_old - (x_old^4 - 4x_old^3 + 5x_old - 1)/(4x_old^3 - 12x_old^2 + 5)
= 1 - (1^4 - 4(1)^3 + 5(1) - 1)/(4(1)^3 - 12(1)^2 + 5)
= 1 - (1 - 4 + 5 - 1)/(4 - 12 + 5)
= 1 - 1/(-3)
= 1 + 1/3
= 4/3
≈ 1.333
Now we have an updated value for x: x_new = 4/3 ≈ 1.333
Step 3: Repeat the process until the value of x_new converges to the actual zero, or until you reach the desired accuracy.
Applying the Newton-Raphson formula again:
x_new = 1.333 - (1.333^4 - 4(1.333)^3 + 5(1.333) - 1)/(4(1.333)^3 - 12(1.333)^2 + 5)
= 1.333 - (3.57742 - 9.18518 + 8.32835 - 1)/(10 - 17.7778 + 5)
≈ 1.334662
Repeat this process until you reach the desired accuracy. Keep updating x_new until it no longer changes significantly.
Using this method, you can estimate the zeros of the polynomial p(x) = x^4 - 4x^3 + 5x - 1.