Step-by-step explanation:
Using the Newton-Raphson method with an initial guess of x0 = 1, we can iteratively approximate a real root of the function f(x) = 2.5x^4 + 3x^3 - 2.6x^2 - 5.1x - 5.6:
1. Start with x0 = 1.
2. Calculate f(x1):
f(x1) = 2.5x1^4 + 3x1^3 - 2.6x1^2 - 5.1x1 - 5.6
f(1) = 2.5(1)^4 + 3(1)^3 - 2.6(1)^2 - 5.1(1) - 5.6
f(1) = 2.5 + 3 - 2.6 - 5.1 - 5.6
f(1) ≈ -7.7
3. Calculate f'(x1):
f'(x1) = 10x1^3 + 9x1^2 - 5.2x1 - 5.1
f'(1) = 10(1)^3 + 9(1)^2 - 5.2(1) - 5.1
f'(1) = 10 + 9 - 5.2 - 5.1
f'(1) ≈ 8.7
4. Update the approximation:
x1 = x0 - f(x0) / f'(x0)
x1 = 1 - (-7.7) / 8.7
x1 ≈ 1.88506
5. Repeat steps 2 to 4 with the new approximation:
f(x2) ≈ -1.087
f'(x2) ≈ 4.884
x2 ≈ 2.124
6. Continue this process until you reach a sufficiently accurate root. In this case, the iterations would continue until you get a more precise estimate of the root. The root appears to be somewhere between x ≈ 2.124 and x ≈ 1.88506.
To find a more accurate root, you can continue these iterations or use a computer algebra system or calculator with built-in root-finding capabilities.