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Question
If α and β are the zeroes of the polynomial f(x) = x2 + px + q, form a polynomial whose zeroes are (α + β)2 and (α − β)2.
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Solution
If α and β are the zeros of the quadratic polynomial f(x) = x2 + px + q
`alpha+beta="-coefficient of x"/("coefficient of "x^2)`
`=(-p)/1`
`alphabeta="constant term"/("coefficient of "x^2)`
`=q/1`
= q
Let S and P denote respectively the sums and product of the zeros of the polynomial whose zeros are (α + β)2 and (α − β)2
S = (α + β)2 + (α − β)2
`S=alpha^2+beta^2+2alphabeta+alpha^2+beta^2-2alphabeta`
`S=2[alpha^2+beta^2]`
`S=2[(alpha+beta)^2-2alphabeta]`
`S=2(p^2-2xxq)`
`S=2(p^2-2q)`
`P=(alpha+beta)^2(alpha-beta)^2`
`P=(alpha^2+beta^2+2alphabeta)(alpha^2+beta^2-2alphabeta)`
`P=((alpha+beta)^2-2alphabeta+2alphabeta)((alpha+beta)^2-2alphabeta-2alphabeta)`
`P=(p)^2((p)^2-4xxq)`
`P=p^2(p^2-4q)`
The required polynomial of f(x) = k(kx2 - Sx + P) is given by
f(x) = k{x2 - 2(p2 - 2q)x + p2(p2 - 4q)}
f(x) = k{x2 - 2(p2 - 2q)x + p2(p2 - 4q)}, where k is any non-zero real number.
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