Answer to Question #252444 in Linear Algebra for lakish

Question

If a third degree polynomial has a lone x-intercept at x=a, discuss what this implies about the linear and quadratic factors of that polynomial.

Answer 1

In general if a polynomial with REAL coefficients have complex roots, then they come in pair and they are each other conjugate.

Thus a 3rd degree polynomial can have either two conjugate complex roots and one real root or three real roots.

Hence a 3rd degree polynomial with REAL coefficients can have either one x-intercept or 3 x-intercepts.

Given that a third degree polynomial has a lone x-intercept at $x=a.$

Let the multiplicity of the root $x=a$ be 1. Then the polynomial has two conjugate complex roots and one real root $x=a.$ The linear factor is $x-a$ and the quadratic factor is $bx^2+cx+d,$ where $c^2-4bd<0.$

Let the multiplicity of the root $x=a$ be 3. Then the polynomial has three equal linear factors and no quadratic factor: $b(x-a)^3.$