Question

If
α & ß are the zeroes of the
polynomial f(x)= 4x²-5x+1, find a
quadratic polynomial whose zeros are α²/β² and β²/α².
​

Answer 1

a and b are zeroes of 4x^2 - 5x +1

a+ b= 5/4

ab = 1/4

( a^2+ b^2) = ( a+b)^2 - 2ab

=25/16 - 1/2

= 25 - 8)/16 = 17/16

Now a quadratic polynomial whose zeroes are a^2/b^2. and b^2/a^2

x^2 - ( a^2/b^2. + b^2/a^2) x + a^2/b^2. b^2/a^2

x^2 - ( a^4 + b^4)/a^2 b^2 )x+ 1

x^2 - ((a^2 + b^2)^2 - 2a^2b^2)/a^2b^2)x + 1

x^2 - ( (17/16)^2 - 2/16. )/ 1/16.)x + 1

x^2 - (16( 17/16)^2 + 16×2/16.)x + 1

x^2 - ( 17^2/16 + 1/2). x. +1

x^2 - ( 289/16. + 1/2) x +1

x^2 - ( 289 + 8)/16. x +1

x^2 - 297/16. x +1

Also its prpportional

16 x^2 - 297x +16