Question

please show all work

2. [10 pts| Use the expansion method to find O(e²) approximations for each of the three roots of x³ + €x² + 1 = 0. ex? Note. For this problem it's useful to make use of the identity eir = –1.

Answer 1

Solution: Considel x² + x + 100 Assume x= xot Exit o(er) then becomes (xo + Exit olen)] + E (not Ex, the 1] + 1 = 0 xo3 + 3 xe x, et der) + er et olen +12 thus x3 + 120 € and 3%82, + 200 - from ③ x 3 =2) k =0, atese 2 e 42 ITE) thus 26 in thus to a ell then from are thus three roots x = 1/3 - t e-c and & 3 - 2