**Q) **If 𝛼, β are zeroes of quadratic polynomial x^{2} – 2x + 3, find the polynomial whose roots are:

1. 𝛼 + 2, 𝛽 + 2

2.

**Ans: **Given polynomial equation x^{2} – 2x + 3 = 0

Comparing with standard polynomial, ax^{2} + 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) = x^{2 }– (sum of the zeroes)x + (product of the zeroes)

= x^{2 }– (6) x + (11)**Hence, the required quadratic polynomial is f(x) = x ^{2 }– 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) = x^{2 }– (sum of the zeroes)x + (product of the zeroes)

= x^{2 }– x +

**Hence, the required quadratic polynomial is f(x) = x ^{2 }– x + **