The given polynomial = x3 – 3x2 + x + 1 and its roots are (a – b), a and (a + b).
Comparing the given polynomial with Ax3 + Bx2 + Cx + D,
we have:
A = 1, B = -3, C = 1 and D = 1
Now, (a – b) + a + (a + b) = \(\frac{-B}A\)
⇒ 3 a = – \(\frac{-3}1\)
⇒ a = 1
Also, (a – b) × a × (a + b) = \(\frac{-D}A\)
⇒ a (a2 – b2 ) = \(\frac{-1}1\)
⇒ 1 (12 – b2 ) = -1
⇒ 1– b2 = -1
⇒ b2 = 2
⇒ b = ±√2
∴ a = 1 and b = ±√2