The bisection method is applied to compute a zero of the function $f(x) = x^4 - x^3 - x^2 - 4$ in the interval $[1,9]$. The method converges to a solution after ______ iterations.

We are given the function $f(x) = x^4 - x^3 - x^2 - 4$ on the interval $[a_1, b_1] = [1, 9]$.

• Evaluate at endpoints:

  $$f(1) = 1^4 - 1^3 - 1^2 - 4 = -4 < 0$$

  $$f(9) = 9^4 - 9^3 - 9^2 - 4 = 6561 - 729 - 81 - 4 = 5747 > 0$$

Iteration 1:

• Midpoint $c_1 = \frac{1 + 9}{2} = 5$

• $f(5) = 5^4 - 5^3 - 5^2 - 4 = 625 - 125 - 25 - 4 = 471 > 0$

• Since $f(1) < 0$ and $f(5) > 0$, the new interval is $[1, 5]$.

Iteration 2:

• Midpoint $c_2 = \frac{1 + 5}{2} = 3$

• $f(3) = 3^4 - 3^3 - 3^2 - 4 = 81 - 27 - 9 - 4 = 41 > 0$

• Since $f(1) < 0$ and $f(3) > 0$, the new interval is $[1, 3]$.

Iteration 3:

• Midpoint $c_3 = \frac{1 + 3}{2} = 2$

• $f(2) = 2^4 - 2^3 - 2^2 - 4 = 16 - 8 - 4 - 4 = 0$

Since $f(2) = 0$, the exact zero is found at the midpoint of the 3rd iteration. Thus, the method converges to a solution after 3 iterations (Option B).