Find the square root of x4 - 4x3 + 8x2 - 8x + 4.
The given expression being of degree four, its square root will be a quadratic expression. Therefore we may assume
x4 - 4x3 + 8x2 - 8x + 4 = (x + lx + m)2, where l and m are to be determined.
(x + lx + m)2 = (x + lx + m) (x + lx + m)
= x4 + lx3 + mx2 + lx3 + l2x2 + lmx + mx2 + lmx + m2
= x4 + 2lx3 + 2mx2 + l2x2 + 2lmx + m2
= x4 + 2lx3 + (l2 + 2m)x2 + 2lmx + m2
x4 - 4x3 + 8x2 - 8x + 4 = x4 + 2lx3 + (l2 + 2m)x2 + 2lmx + m2
Comparing the coefficients of powers of x on either sides, we get
2l = -4, l2 + 2m = 8, 2lm = -8, m2 = 4
Therefore, l = -2 and m = 2.
Therefore, the square root is x2 + lx + m = x2 - 2x + 2.
Directions: Solve the following problems using above method. Also write at least 5 examples of your own.