Question

1. The polynomial f(x) = 0.0074.0+ - 0.284.2 + 3.35522 - 12.183.c +5 has a real root between 15 and 20. Use a bracketing method to find the solution accurate up to four significant digits.

Answer 1

Given :

f(x) = 0.0074 x⁴ - 0.284 x³ + 3.355 x² - 12.183 x + 5 on (15,20)

Solution :

By Using Bisection method procedure under bracketing method, we can easily solve this polynomial function.

Bracketing Bisection Method Method Procedures Step 1: Take a = a; bo=b . . Step 2: For n=1,2,... Continue till satisfies, (6) If / 8 (anut b) =o) then & - anut bune where &- Root a) f(blama) = (autom 20) then take az any br=anut brot 2 a Unless take ove o n a= ano+bn-i -= bn-bn- 0.3

Sola. :- f(15) = (0.0074x157) -(0.284x15) +(3.355*15) - 112.183 x 15) + 5 = -6.745 f (20) = (0.0074x264) - (0.244x203) + (3.355 x202) - (12.183 x 20) +5 = 15.34. f(15). f (20) = -103.47 <0 i. There is a root for the given function in [15,20]. Set, a = 15, bo = 20. 8 (007b) = /15720 ) = 4(175) = -3.76 Substitute a value as 17.5 +0 - 6.745 X - 8.76=59.09. . I Set a = ao too = 17.5; b. = b = 20 ao=15.0000 bo = 20.0000 a = 17.5000 ; b. = 20.0000

By using these a_n,b_{n values we can obtain the following table.}

f(x) = 0.0074 x4 - 0.284 x3 + 3.355 x2 - 12.183 x +5
Bisection Method
Iteration No	an	bn	+ь. 5, м	f(en)
0	15.0000	20.0000	17.5000	-8.7572
1	17.5000	20.0000	18.7500	-138.1922
2	17.5000	18.7500	18.1250	-73.1413
3	18.1250	18.7500	18.4375	-105.6068

We can continue this iteration till it satisfies condition.