Question

If the zeroes of the polynomial x³ – 3x² + x + 1 are a – b, a, a + b, find a and b.

Answer 1

Given polynomial x³ – 3x² + x + 1 

Since, (a – b), a, (a + b) are the zeroes of the polynomial x³ – 3x² + x + 1. 

Therefore, sum of the zeroes 

= (a – b) + a + (a + b) = \(\frac{-(-3)}{1}\) = 3 

So, 3a = 3 ⇒ a = 1 

∴ Sum of the products of its zeroes taken two at a time. 

= a(a – b) + a(a + b) + (a + b) (a – b) = \(\frac{1}{1}\) = 1 

⇒ a² – ab + a² + ab + a² – b² = 1 

⇒ 3a² – b² = 1 

So, 3(1)² – b² = 1

⇒ 3 – b² = 1 

⇒ b² = 2 

⇒ b = √2 = ± √2 

Here, a = 1 and b = ± √2