Form the polynomial equation whose roots are the squares of the roots

1.	Form the polynomial equation whose roots are the squares of the roots of x3 + 3x2 – 7x + 6 = 0
Answer» Given equation is f(x) = x³ + 3x² – 7x + 6 = 0 Required equation is f(√x) = 0 ⇒ x√x + 3x – 7√x + 6 = 0 ⇒ √x(x – 7) = – (3x + 6) Squaring on both sides, ⇒ x(x – 7)² = (3x + 6)² ⇒ x(x²– 14x + 49) = 9x² + 36 +36x ⇒ x³ – 14x² + 49x – 9x² – 36x – 36 = 0 i.e. x³ – 23x² + 13x – 36 = 0

Form the polynomial equation whose roots are the squares of the roots of x3 + 3x2 – 7x + 6 = 0

Answer»

Given equation is f(x) = x³ + 3x² – 7x + 6 = 0

Required equation is f(√x) = 0

⇒ x√x + 3x – 7√x + 6 = 0

⇒ √x(x – 7) = – (3x + 6)

Squaring on both sides,

⇒ x(x – 7)² = (3x + 6)²

⇒ x(x²– 14x + 49) = 9x² + 36 +36x

⇒ x³ – 14x² + 49x – 9x² – 36x – 36 = 0

i.e. x³ – 23x² + 13x – 36 = 0

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Form the polynomial equation whose roots are the squares of the roots of x3 + 3x2 – 7x + 6 = 0

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