Correct Answer - Option 1 : [a
2x
2 – (b
2 – 2ac)x + c
2]/a
2
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