Question

if a, b and c are roots of the equation x3 + qx + r is equals to zero, give a polynomial whose roots are b2c2, c2a2, a2b2 and whose coefficients are given in terms of q and r​

Answer 1

x³ - q² x² - 2qr² x - r⁴ = 0

Step-by-step explanation:

The given equation is

x^3 + qx + r = 0

Since a, b, c are the roots of the above equation, using the relation between roots and coefficients, we get

a + b + c = 0 ..... (1)

ab + bc + ca = q ..... (2)

abc = - r ..... (3)

We have to find the equation whose roots are b²c², c²a², a²b².

Now,

b²c² + c²a² + a²b²

= (bc + ca + ab)² - 2 (bc ca + ca ab + ab bc)

= (bc + ca + ab)² - 2 abc (a + b + c)

= q² - 2 (- r) . 0 [ by (1), (2) & (3) ]

= q²,

b²c² c²a² + c²a² a²b² + a²b² b²c²

= a²b²c² (a² + b² + c²)

= (abc)² {(a + b + c)² - 2 (ab + bc + ca)}

= (- r)² (0² - 2q) [ by (1), (2) & (3) ]

= - 2qr² and

b²c² c²a² a²b²

= (abc)⁴

= (- r)⁴ [ by (3) ]

= r⁴

∴ the required equation with roots b²c², c²a², a²b² is given by

x³ - (b²c² + c²a² + a²b²) x² + (b²c² c²a² + c²a² a²b² + a²b² b²c²) x - b²c² c²a² a²b² = 0

or, x³ - q² x² - 2qr² x - r⁴ = 0