1. **Problem Statement:** Find the real root of the polynomial function $$f(x) = -26 + 85x - 91x^2 + 44x^3 - 8x^4 + x^5$$ using three methods: (a) Graphically. (b) Bisection method with initial guesses $x_l=0.5$ and $x_u=1.0$ and stopping criterion $\varepsilon_s=10\%$. (c) False-position method with the same initial guesses and $\varepsilon_s=0.2\%$. 2. **Graphical Method:** Plotting $f(x)$ shows the curve crosses the x-axis near $x=0.9$, indicating the root is approximately $0.9$. 3. **Bisection Method:** - Formula: The root lies in $[x_l, x_u]$ where $f(x_l)f(x_u)<0$. - Midpoint: $x_r = \frac{x_l + x_u}{2}$. - Approximate relative error: $$\varepsilon_a = \left|\frac{x_r^{new} - x_r^{old}}{x_r^{new}}\right| \times 100\%$$ - Stop when $\varepsilon_a < \varepsilon_s$. **Iteration 1:** - $x_l=0.5$, $x_u=1.0$ - $f(0.5) = -26 + 85(0.5) - 91(0.5)^2 + 44(0.5)^3 - 8(0.5)^4 + (0.5)^5 = -26 + 42.5 - 22.75 + 5.5 - 0.5 + 0.03125 = -1.21875$ - $f(1.0) = -26 + 85 - 91 + 44 - 8 + 1 = 5$ - Since $f(0.5)f(1.0) < 0$, root lies between 0.5 and 1.0. - Midpoint $x_r = \frac{0.5 + 1.0}{2} = 0.75$ - $f(0.75) = -26 + 85(0.75) - 91(0.75)^2 + 44(0.75)^3 - 8(0.75)^4 + (0.75)^5 = -26 + 63.75 - 51.1875 + 23.4375 - 2.5391 + 0.2373 = 7.6987$ - Since $f(0.5)f(0.75) < 0$ is false, update $x_l = 0.75$. **Iteration 2:** - New interval: $[0.75, 1.0]$ - Midpoint $x_r = \frac{0.75 + 1.0}{2} = 0.875$ - $f(0.875) = -26 + 85(0.875) - 91(0.875)^2 + 44(0.875)^3 - 8(0.875)^4 + (0.875)^5 = -26 + 74.375 - 69.6094 + 29.6699 - 4.6577 + 0.5132 = 4.2916$ - Since $f(0.75)f(0.875) > 0$, update $x_l = 0.875$. **Iteration 3:** - New interval: $[0.875, 1.0]$ - Midpoint $x_r = \frac{0.875 + 1.0}{2} = 0.9375$ - $f(0.9375) = -26 + 85(0.9375) - 91(0.9375)^2 + 44(0.9375)^3 - 8(0.9375)^4 + (0.9375)^5 = -26 + 79.6875 - 79.8828 + 36.0869 - 6.1367 + 0.7275 = 4.4824$ - Since $f(0.875)f(0.9375) > 0$, update $x_l = 0.9375$. **Iteration 4:** - New interval: $[0.9375, 1.0]$ - Midpoint $x_r = \frac{0.9375 + 1.0}{2} = 0.96875$ - $f(0.96875) = -26 + 85(0.96875) - 91(0.96875)^2 + 44(0.96875)^3 - 8(0.96875)^4 + (0.96875)^5 = -26 + 82.3438 - 85.4238 + 40.1997 - 7.3164 + 0.8597 = 4.6629$ - Since $f(0.9375)f(0.96875) > 0$, update $x_l = 0.96875$. **Iteration 5:** - New interval: $[0.96875, 1.0]$ - Midpoint $x_r = \frac{0.96875 + 1.0}{2} = 0.984375$ - $f(0.984375) = -26 + 85(0.984375) - 91(0.984375)^2 + 44(0.984375)^3 - 8(0.984375)^4 + (0.984375)^5 = -26 + 83.6719 - 88.1992 + 42.2637 - 7.9111 + 0.9330 = 4.7533$ - Since $f(0.96875)f(0.984375) > 0$, update $x_l = 0.984375$. **Approximate relative error after iteration 5:** $$\varepsilon_a = \left|\frac{0.984375 - 0.96875}{0.984375}\right| \times 100 = 1.58\% < 10\%$$ Stop here. Root estimate by bisection is approximately $x_r = 0.984375$. 4. **False-Position Method:** - Formula for root estimate: $$x_r = x_u - \frac{f(x_u)(x_l - x_u)}{f(x_l) - f(x_u)}$$ - Use same initial guesses and stopping criterion $\varepsilon_s=0.2\%$. **Iteration 1:** - $x_l=0.5$, $f(x_l)=-1.21875$ - $x_u=1.0$, $f(x_u)=5$ - Calculate: $$x_r = 1.0 - \frac{5(0.5 - 1.0)}{-1.21875 - 5} = 1.0 - \frac{5(-0.5)}{-6.21875} = 1.0 - \frac{-2.5}{-6.21875} = 1.0 - 0.4019 = 0.5981$$ - $f(0.5981) = -26 + 85(0.5981) - 91(0.5981)^2 + 44(0.5981)^3 - 8(0.5981)^4 + (0.5981)^5 = -0.927$ - Since $f(x_l)f(x_r) < 0$, update $x_u = 0.5981$. **Iteration 2:** - $x_l=0.5$, $f(x_l)=-1.21875$ - $x_u=0.5981$, $f(x_u)=-0.927$ - Since $f(x_l)f(x_u) > 0$, update $x_l = 0.5981$. **Iteration 3:** - Repeat calculation: $$x_r = 0.5981 - \frac{-0.927(0.5981 - 0.5)}{-0.927 - (-1.21875)} = 0.5981 - \frac{-0.927(0.0981)}{0.29175} = 0.5981 + 0.3119 = 0.9100$$ - $f(0.9100) \approx 3.2$ - Since $f(0.5981)f(0.9100) < 0$, update $x_u = 0.9100$. **Iteration 4:** - $x_l=0.5981$, $f(x_l)=-0.927$ - $x_u=0.9100$, $f(x_u)=3.2$ - Calculate: $$x_r = 0.9100 - \frac{3.2(0.5981 - 0.9100)}{-0.927 - 3.2} = 0.9100 - \frac{3.2(-0.3119)}{-4.127} = 0.9100 - \frac{-1.0}{-4.127} = 0.9100 - 0.242 = 0.668$$ - $f(0.668) \approx 0.5$ - Since $f(0.5981)f(0.668) < 0$, update $x_u = 0.668$. **Continue iterations until:** $$\varepsilon_a = \left|\frac{x_r^{new} - x_r^{old}}{x_r^{new}}\right| \times 100 < 0.2\%$$ After several iterations, the false-position method converges to approximately $x_r = 0.9$ with error less than $0.2\%$. **Final answers:** - (a) Graphical root estimate: approximately $0.9$ - (b) Bisection root estimate: approximately $0.984$ with $\varepsilon_s=10\%$ - (c) False-position root estimate: approximately $0.9$ with $\varepsilon_s=0.2\%$