Question

If alpha and bita are the zeroes of the quardratic polynomial f(x)=xsquare+px+q, form a polynomial whose zeroes are (alpha+bita) whole square and (alpha-bita)whole square

Answer 1

alpha =alpha +beta whole square
beta=alpha-beta whole square
alpha+beta=(a^2+2ab+b^2)

Answer 2

Given polynomial,
f(x)=x²+px+q
Let 'A' and 'B' be the zeroes.
Now we have,
Sum of the zeroes=A+B=-p
A+B=-p..........................................1

Product of zeroes=A×B=q
AB=q..............................................2
Now as per given,
(A+B)² and (A-B)² are zeroes of the required polynomial.
(A+B)²=(-p)²=p².............................3

(A-B)²=A²+B²-2AB
(A-B)²=(A+B)²-4AB
(A-B)²=(-p)²-4(q)
(A-B)²=p²-4q..................................4
We know that any polynomial is in the form of,
=k{x²-(sum of the zeroes)x+product of zeroes}
By eq3 and eq4 we get,
=k{x²-(p²+p²-4q)x+(p²)(p²-4q)}

=k{x²-(2p²-4q)x+p²(p²-4q)}

=x²-2x(p²-2q)+p²(p²-4q)

Hence x²-2x(p²-2q)+p²(p²-4q) is the required polynomial.