If $α$ and $β$ are the zeros of the quadratic polynomial $f (x) = x^{2} - p (x +, 1) - c$ , show that $(α + 1) (β + 1) = 1 - c .$

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Solution

$f (x) = x^{2} - p (x + 1) - c$

$f (x) = x^{2} - p x - (p + c)$

Roots are $α, β$

$α + β = p \dots (1)$

$α β = - (p + c) \dots (2)$

$(α + 1) (β + 1) = α β + α + β + 1$

$= - (p + c) + p + 1 \dots (s u b s t i t u t i n g v a l u e s f r o m (1) a n d (2))$

$(α + 1) (β + 1) = 1 - c$

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