Cubic Polynomials

We have already learned about the relationship between the zeroes and coefficients of linear and quadratic polynomials. Now, let's explore whether a similar relationship holds for cubic polynomials.

Example: A Cubic Polynomial

Consider the polynomial: p(x) = 2x3 - 5x2 - 14x + 8

We are given that its zeroes (roots) are: x = 4, -2, 12.

Since a cubic polynomial can have at most threetwofour zeroes, these are all the roots of p(x).

2x3 - 5x2 - 14x + 8

1. Sum of the Zeroes

= 4 + (-2) + 12 = 5225

= −−52 = −coefficient ofx2/coefficient ofx3

2. Product of its Zeroes

= 4 × (-2) × 12 = = −8228

= −constant term/coefficient ofx3

However,there is one more relationship here.Consider the sum of the products of the zeros taken at a time. We have :

= {4 × (-2)} + {(-2) × 12} + {12 × 4}

= - 8 - 1 + 2 = =

In general, it can be proved that if α, β, γ are the zeroes of the cubic polynomial:

ax3 + bx2 + cx + d

α + β + γ = −ba

αβ + βγ + γα = cd

and αβγ = −da

Continue