Answer to Question #251134 in Calculus for bri

Question

2. 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

Since a third degree polynomial $f$ has a lone $x$-intercept at $x=a$, it has a unique root $x=a.$ If the multiplicity of this root is 3, then $f(x)=c(x-a)^3,$ and in this case the polynomial has three (the same) linear factors. If the root $x=a$ is of multiplicity 2, then the third factor is linear, and polinomial has the root $x=b\ne a,$ which is impossible according to uniqueness of a root. If the multiplicity of this root is 1, then $f(x)=(x-a)(bx^2+cx+d),$ and polynomial $g(x)=bx^2+cx+d$ has no roots. It follows that the last polynomial is irreducible, and hence it is the quadratic factor of polynomial $f$.