Answer to Question #253250 in Functional Analysis for Toya

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

Solution.

Since a third degree polynomial f has a lone x-intercept at x=a, it has a unique root x=a.

1) If the multiplicity of this root is 3, then

$$f(x)=k(x-a)^3$$

and in this case the polynomial has three linear factors.

2) If the root x=a is of multiplicity 2, then the third factor is linear, and polinomial has the root $x=b\neq a$

which is impossible according to uniqueness of a root.

3) If the multiplicity of this root is 1, then

$$f(x)=(x-a)(mx^2+nx+p)$$

and polynomial $mx^2+nx+p$ has no roots. It follows that the last polynomial is irreducible, and hence it is the quadratic factor of polynomial f.