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.
Answers
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)).
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...✌