Question

if a and b are zeroes of the polynomial x^2-x-6, then find a quadratic polynomial whose zeroes are (3a+2b) and (2a+3b).​

Answer 1

Good Morning!

a + b = 1

And

a×b = -6

Required quadratic polynomial in y is given by p(y) = y² - ( S.O.Z )y + ( P.O.Z )

S.O.Z = ( 3a + 2a + 3b + 2b )

S.O.Z = { a + 2(a + b) + b + 2(a + b) }

S.O.Z ={ ( a + b ) + 2 ( a + b ) + 2 ( a + b )}

S.O.Z = { 1 + 2 + 2 }

S.O.Z = 5

And

P.O.Z = { 6a² + 9ab + 4ab + 6b² }

P.O.Z = { 6( a² + b² ) + 13ab }

P.O.Z = { 6 ( a + b )² -2ab + 13ab }

P.O.Z = { 6 ( a + b )² + 11ab }

P.O.Z = { 6 -66 }

P.O.Z = -60

So, Required Quadratic polynomial is

p(y) = y² - 5y - 60

Note:-

Here S.O.Z means Sum of zeros. And P.O.Z means product of zeros.