If a and ? are the zeroes of the quadratic polynomial f(x) = x ^ 2 + x - 2 then find a polynomial whose zeroes are 2alpha + 1 and 2beta + 1

Solution: 

Given: α and β are zeroes of the quadratic polynomial f(x) = x2 + x − 2.

To find zeroes: f(x) = 0

x2 + x − 2 = 0
x2 + 2x − x − 2 = 0
⇒ x(x + 2) − 1(x + 2) = 0
⇒ (x + 2)(− 1) = 0
⇒ x = 1 or −2
∴ α = 1 and β = −2
⇒ 2α + 1​ = 3 and ​​2β + 1 = −3
Thus, the equation of polynomial with zeros 3 and −3 is given by:

P(x) = (+ 3)(x − 3) = x2 − 9                        [∵ (a + b)(a − b) = a2 − b2]

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