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
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
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
which is impossible according to uniqueness of a root.
3) If the multiplicity of this root is 1, then
and polynomial has no roots. It follows that the last polynomial is irreducible, and hence it is the quadratic factor of polynomial f.
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