Q)  If 𝛼, β are zeroes of quadratic polynomial x² – 2x + 3, find the polynomial whose roots are:
1. 𝛼 + 2, 𝛽 + 2
2.

Ans: Given polynomial equation x² – 2x + 3 = 0

Comparing with standard polynomial, ax² + b x + c = 0, we get,

a =  1, b = – 2, c = 3

Since, its given that the roots of the polynomial be α and β.

and we know that sum of roots (α + β) = 

   α + β =  = 2  …………… (i)

Also, we know that the product of the roots (α x β) =

α . β = = 3 …………. (ii)

Next, Let’s find the polynomials:

(1) Polynomial for roots (𝛼 + 2, 𝛽 + 2):

∵ Sum of the zeroes of new polynomial = (α + 2) + (β + 2) = (α + β) + 4

By transferring values from equations (i), we get:

∴ Sum of the zeroes of new polynomial = (2) + 4 = 6

Next, Product of the zeroes of new polynomial = (α + 2)(β + 2)

= αβ + 2α + 2β + 4

= αβ + 2(α + β) + 4

∴ Product of the zeroes of new polynomial = (3) + 2 (2) + 4 = 11

Since, quadratic polynomial f(x) = x² – (sum of the zeroes)x + (product of the zeroes)
= x² – (6) x + (11)
Hence, the required quadratic polynomial is f(x) = x² – 6x + 11

(2) Polynomial for roots ()

∵ Sum of the zeroes of new polynomial =

=

=

=

By transferring values from equations (i) and (ii), we get:

∴ Sum of the zeroes of new polynomial =  =

∴ Sum of the zeroes of new polynomial =

Next, Product of the zeroes of new polynomial =

=

=

=

By transferring values from equations (i) and (ii), we get:

Product of the zeroes of new polynomial =  =

∴ Product of the zeroes of new polynomial =   

Since, quadratic polynomial f(x) = x² – (sum of the zeroes)x + (product of the zeroes)

= x² – x +

Hence, the required quadratic polynomial is f(x) = x² – x +