If the zeros of the quadratic polynomial x2 + (a + 1)x + b are 2 and - 3,
Then
Zeroes of a polynomial is all the values of x at which the polynomial is equal to zero.
2 and - 3 are the zeroes of the polynomial p(x) = x2 + (a + 1)x + b
i.e. p(2) = 0 and p(- 3) = 0
p(2) = (2)2 + (a + 1)(2) + b = 0
= 4 + 2a + 2 + b = 0
= 6 + 2a + b = 0 (1)
P(- 3) = (- 3)2 + 9 + (a + 1)(- 3) + b = 0
= 9 - 3a - 3 + b = 0
= 6 - 3a + b = 0 (2)
Equating (1) = (2), as both the equations are equal to zero. Hence both equations are equal to each other.
6 + 2a + b = 6 - 3a + b
= 5a = 0
⇒ a = 0
Putting the value of ‘a’ in (1)
6 + 2(0) + b = 0
⇒ b = - 6
OR
The equation of a quadratic polynomial is given by x2 - (sum of the zeroes) x + (product of the zeroes)
Sum of the zeroes = - (coefficient of x) ÷ coefficient of x2
⇒ sum of the zeroes = - (a + 1)
= 2 - 3 = - a - 1
= - 1 + 1 = - a
= - a = 0
⇒ a = 0
Product of the zeroes = constant term ÷ coefficient of x2
⇒ b = product of the zeroes
= 2(- 3)
= 6