Question #253250

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


1
Expert's answer
2021-10-20T00:54:02-0400

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(xa)3f(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=bax=b\neq a

which is impossible according to uniqueness of a root.

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


f(x)=(xa)(mx2+nx+p)f(x)=(x-a)(mx^2+nx+p)

and polynomial mx2+nx+pmx^2+nx+p 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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