To approximate the real number solutions of the polynomial function f(x) = x³ – 2x² + 5x – 6, we can use several methods, including graphing, the intermediate value theorem, or numerical methods like the Newton-Raphson method. Here’s a step-by-step guide:
Step 1: Analyze the Function
First, let’s evaluate the function at a few points to understand its behavior. We want to know where the function changes signs, as this indicates the presence of a root (a real number solution).
Evaluate f(x) at Various Points
- f(1): f(1) = 1³ – 2(1)² + 5(1) – 6 = 1 – 2 + 5 – 6 = -2
- f(2): f(2) = 2³ – 2(2)² + 5(2) – 6 = 8 – 8 + 10 – 6 = 4
- f(3): f(3) = 3³ – 2(3)² + 5(3) – 6 = 27 – 18 + 15 – 6 = 18
- f(0): f(0) = 0³ – 2(0)² + 5(0) – 6 = -6
From these evaluations, we see that:
- f(1) = -2 → negative
- f(2) = 4 → positive
Thus, by the intermediate value theorem, there is at least one real root between x = 1 and x = 2.
Step 2: Use the Bisection Method
To narrow down the root, we can apply the bisection method:
- Set a = 1 and b = 2.
- Compute c = (a + b) / 2 = 1.5.
- Evaluate f(1.5): f(1.5) = (1.5)³ – 2(1.5)² + 5(1.5) – 6 = 3.375 – 4.5 + 7.5 – 6 = 0.375
- Update bounds:
- Since f(1) < 0 and f(1.5) > 0, the root lies in [1, 1.5].
- Continue bisecting until reaching an acceptable precision.
Step 3: Apply Newton-Raphson Method
If you prefer a faster convergence, use the Newton-Raphson method:
- Start with an initial guess (from the results of the bisection method).
- Iterate using the formula: x_{n+1} = x_n – f(x_n) / f'(x_n), where f'(x) = 3x² – 4x + 5 is the derivative.
Conclusion
Through these methods, especially by evaluating values in the target range and using numerical methods like the bisection or Newton-Raphson, you can effectively approximate the real number solutions to the polynomial function f(x) = x³ – 2x² + 5x – 6. Always remember to check for multiple roots, as polynomial equations can have multiple solutions within any given range.