Question

If α and β are the zeroes of the quadratic polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α – β)2.

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Answer 1

Solution:

We have,

f(x) = x2 + px + q

Sum of the zeroes = α + β = -p

Product of the zeroes = αβ = q

From the question,

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

= (α + β)2 + α2 + β2 – 2αβ

= (α + β)2 + (α + β)2 – 2αβ – 2αβ

= (- p)2 + (- p)2 – 2 × q – 2 × q

= p2 + p2 – 4q

= p2 – 4q

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

= (- p)2((- p)2 - 4q)

= p2 (p2–4q)

So, the quadratic polynomial is,

x2 – (sum of the zeroes)x + (product of the zeroes)

= x2 – (2p2 – 4q)x + p2(p2 – 4q)

Hence, the required quadratic polynomial is f(x) = k(x2 – (2p2 –4q) x + p2(p2 - 4q)).

Answer 2

Heyaa☺

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Given, polynomial f(x) = x2 + px + q

Let α and β are the zeros of the polynomial

Now, α + β = -p

=> (α + β)2 = (-p)2

=> (α + β)2 = p2

and α * β = q

Again (α - β)2 = (α + β)2 - 4 * α * β

=> (α - β)2 = (-p)2 - 4 * q

=> (α - β)2 = p2 - 4q

Now, the polynomial whose zeroes are (α + β)2 and (α - β)2 is

f(x) = x2 - {sum of zeros}x + product of zeros

=> f(x) = x2 - {(α + β)2 + (α - β)2 }x + (α + β)2 * (α - β)2

=> f(x) = x2 - {p2 + p2 - 4q}x + (p2 - 4q) * p2

=> f(x) = x2 - (2p2 - 4q)x + p4 - 4 * p2 * q

=> f(x) = x2 - (2p2 - 4q)x + p4 - 4 qp2

This is the required polynomial.

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Hope it helps...✌