Question

Find the zeros of the polynomial x² + x – p(p + 1).

Answer 1

f(x) = x² + x – p(p + 1)

By adding and subtracting px, we get

f(x) = x² + px + x – px – p(p + 1)

= x² + (p + 1)x – px – p(p + 1)

= x[x + (p + 1)] – p[x + (p + 1)]

= [x + (p + 1)] (x – p)

f(x) = 0

⇒ [x + (p + 1)] (x – p) = 0

⇒ [x + (p + 1)] = 0 or (x – p) = 0

⇒ x = – (p + 1) or x = p

So, the zeroes of f(x) are – (p + 1) and p.