Question

If the zeros of the polynomial x³ – 3x² + x + 1 are (a – b), a and (a + b), find the values of a and b.

Answer 1

Given: The zeros of the polynomial x³ – 3x² + x + 1 are (a – b), a and (a + b).

Step-wise calculation:

1. Sum of zeros = (a – b) + a + (a + b) = 3a. 

For the cubic x³ – 3x² + x + 1 (monic), 

Sum of zeros = –(coefficient of x²) 

= –(–3) 

= 3

Therefore, 3a = 3 ⇒ a = 1.

2. Product of zeros = (a – b) × a × (a + b) 

= a(a² – b²) 

For a monic cubic the product = –(constant term) = –1.

So, a(a² – b²) = –1. 

Substitute a = 1: 1(1 – b²) = –1 

⇒ 1 – b² = –1 

⇒ b² = 2 

⇒ `b = ±sqrt(2)`

⇒ a = 1 and b = `±sqrt(2)`