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$\alpha$ and $\beta$ are zeroes of a quadratic polynomial $x^2 - ax - b$. Obtain a quadratic polynomial whose zeroes are $3\alpha + 1$ and $3\beta + 1$.
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$\alpha + \beta = a, \alpha\beta = -b$
Sum of zeroes of required polynomial
$= (3\alpha + 1) + (3\beta + 1) = 3(\alpha + \beta) + 2 = 3a + 2$
Product of zeroes of required polynomial
$= (3\alpha + 1)(3\beta + 1) = 9\alpha\beta + 3(\alpha + \beta) + 1 = -9b + 3a + 1$
$\therefore$ The required polynomial is $x^2 - (3a + 2)x + (3a - 9b + 1)$
Sum of zeroes of required polynomial
$= (3\alpha + 1) + (3\beta + 1) = 3(\alpha + \beta) + 2 = 3a + 2$
Product of zeroes of required polynomial
$= (3\alpha + 1)(3\beta + 1) = 9\alpha\beta + 3(\alpha + \beta) + 1 = -9b + 3a + 1$
$\therefore$ The required polynomial is $x^2 - (3a + 2)x + (3a - 9b + 1)$