if alpha and beta are zeros of p(x)= x²-2x+3, find a polynomial whose zeros are alpha-1/alpha+1 and beta-1/beta+1
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? and ? are the roots of x2 + px + q. Therefore,
? + ? = -p and ?? = q
Now, 1/? + 1/? = ?+?/?? = -p/q
1/? . 1/? = 1/?? = 1/q
Thus, the required polynomial is:
x2 - (sum of roots)x + (product of roots)
x2 - (-p/q)x + 1/q or qx2 + px + 1
? + ? = -p and ?? = q
Now, 1/? + 1/? = ?+?/?? = -p/q
1/? . 1/? = 1/?? = 1/q
Thus, the required polynomial is:
x2 - (sum of roots)x + (product of roots)
x2 - (-p/q)x + 1/q or qx2 + px + 1
AKANKSHASHARMA:
(alpha-1\alpha+1) and (beta-1\beta+1) are the zeros
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