Question

If the zeroes of the polynomial f ( x ) = x^3-3x^2+x+1 are a-b, a, a+b, find a and b​

Answer 1

Given that a-b, a, a+b are roots of given polynomial x³-3x²+x+1

From this polynomial,
Sum of the roots ⇒ a-b+a+a+b = -coefficient of x²/ coefficient of x³
                             ⇒ 3a = -(-3)/1 = 3
                                      a = 1   --------- One of the root (1)

Product of roots ⇒ (a-b)(a+b)a = -constant/coefficient of x³
                          ⇒ (a²-b²)a = -1/1
                          ⇒ a³ - ab² = -1       {placing the value a = 1 from (1)}
                          ⇒ 1 - b² = -1
                          ⇒ b² = 2
                          ⇒ b = √2
                          
∴ a = 1, b = √2