P(x) ≡ 4a 0 x 3 + 3a 1 x 2 + 2a 2 x + a 3 is a polynomial and hence is continuous for all x. P(x) = 0 has a root in (–1, 0) if it takes both positive and negative values in (–1, 0) as continuity implies that P(x) = 0 at least one point. This will happen if either P(–1).P(0) < 0 or the area enclosed by the graph of P(x), the x-axis and the ordinates at x = – 1 and x = 0 is zero.

As P(0) = a 3 > 0, P(–1) = – 4a 0 + 3a 1 – 2a 2 + a 3 < 0 or 4a 0 + 2a 2 > 3a 1 + a 3

dx = 0 gives a 0 + a 2 = a 1 + a 3 .