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If the zeroes of polynomial ax2 + bx + c, are α and β then what will be the polynomial having zeroes α2 and β2.


1. [a2x2 – (b2 – 2ac)x + c2]/a2
2. [a2x2 – (b2 + 2ac)x + c2]/a3. [a2x2 + (b2 – 2ac)x + c2]/b4. [a2x2 – (b2 – 2ac)x + c2]/b<span style="posi

1 Answer

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Correct Answer - Option 1 : [a2x2 – (b2 – 2ac)x + c2]/a2

Given:

The polynomial is ax2 + bx + c

Zeroes of polynomial are α and β

Concept used:

For a given polynomial ax2 + bx + c

Sum of zeroes = –b/a

Product of zeroes = c/a

And the polynomial can be written as x2 – Sx + P

Here, S is sum of zeroes and P is product of zeroes

Solution:

Sum of zeroes = α + β = –b/a

Squaring both sides we’ll get

(α + β)2 = (b/a)2

⇒ α2 + β2 + 2αβ = b2/a2      ---- (1)

Product of zeroes = α × β = c/a

⇒ αβ = c/a      ---- (2)

Solving (1) and (2) we’ll get

α2 + β2 + 2c/a = b2/a2

⇒ α2 + β2 = b2/a2 – 2c/a

⇒ α2 + β2 = (b2 – 2ac)/a2

As the zeroes of new polynomials are α2 and β2

Sum of zeroes (S) = (b2 – 2ac)/a2

Product of zeroes (P) = c2/a2

The new polynomial will be x2 – (b2 – 2ac)x/a2 + c2/a2

∴ The polynomial will be [a2x2 – (b2 – 2ac)x + c2]/a2

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