Question

If alpha and beta are the zeroes of the polynomial f(x)=x2- p(x+1) - c show that (alpha+1) (Beta +1) = 1-c

Answer 1

see explanation

Explanation:

Given

f(x) = x² - p(x + 1) - c

= x² - px - p - c ← in standard form

with a = 1, b = - p and c = - p - c

Given that α and β are the zeros of f(x), then

α + β = -
`(b)/(a)` and αβ =
`(c)/(a)`, thus

α + β = -
`(-p)/(1)` = p , and

αβ =
`\frac{-p-c}1}` = - p - c

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(α + 1)(β + 1) ← expand factors

= αβ +α + β + 1 ← substitute values from above

= - p - c + p + 1

= - c + 1 = 1 - c ← as required