**Q) **If 𝛼 and β are zeroes of a polynomial 6 x^{2} – 5 x + 1, then form a quadratic polynomial whose zeroes are 𝛼^{2} and 𝛽^{2} .

**Ans: **

**Step 1:** Given polynomial equation 6 x^{2} – 5 x + 1 = 0

Comparing with standard polynomial, ax^{2} + b x + c = 0, we get,

a = 6, b = – 5, c = 1

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

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

∴ α + β = …………… (i)

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

∴ α . β = …………. (ii)

**Step 2:** The zeroes for new polynomial given as (𝛼^{2} , 𝛽^{2} ):

∵ Sum of the zeroes of new polynomial = (α^{2 }+ β^{2} ) = (α + β)^{2} – 2 α . β

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

∴ Sum of the zeroes of new polynomial =

Next, Product of the zeroes of new polynomial = (α^{2}).(β^{2}) = (α . β)^{2}

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

∴ Product of the zeroes of new polynomial =

**Step 3:** New quadratic polynomial f(x) = x^{2 }– (sum of the zeroes) x + (product of the zeroes)

∴

at f(x) = 0, polynomial is ∴

∴ 36 X^{2} – 13 X + 1 = 0

**Hence, the required quadratic polynomial is 36 X ^{2 }– 13 X + 1 = 0.**

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