Answer to Question #250128 in Calculus for shantel

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

Let’s walk through the possibilities. The existence of the single x-intercept means there must be at least one linear factor so we’ll start from there.

1 real root and 2 complex roots This would give you one linear factor and one quadratic factor.

2 real roots and 1 complex root This case can’t happen because the Complex Conjugate Root theorem says the complex roots have to come in pairs.

3 real roots This would give you three linear factors and is a possible result. The polynomial would have to have the form (x−a)³.

So you would end up with two possibilities: either one linear factor and one quadratic factor or three, identical linear factors.