Question
(A) Let \(f(x)=\frac{p(x)}{q(x)}\) , where p and q are both fourth-degree polynomial functions. Discuss the possible number of discontinuities, removable and nonremovable.
(B) How does the situation change if q is a fifth-degree polynomial?
(C) What is the situation regarding discontinuities if p is a fifth-degree polynomial, but q is a fourth-degree polynomial?
▶️Answer/Explanation
(A) Polynomials of even degree do not necessarily have a root, so f (x) need not have any discontinuities. Also, f (x) may have as many as four discontinuities. These continuities will only be removable where p and q have roots in common.
(B) Since polynomials of odd degree will have at least one root, f (x) will have at least one discontinuity, and as many as five. But the polynomial need not have any nonremovable discontinuities due to multiplicity of roots.
(C) This situation is the same as the first. Roots of the numerator do not produce discontinuities.
Question
(A) Let \(f(x)=\frac{p(x)}{q(x)}\) , where p and q are both fourth-degree polynomial functions. Discuss the possible number of discontinuities, removable and nonremovable.
(B) How does the situation change if q is a fifth-degree polynomial?
(C) What is the situation regarding discontinuities if p is a fifth-degree polynomial, but q is a fourth-degree polynomial?
▶️Answer/Explanation
(A) Polynomials of even degree do not necessarily have a root, so f (x) need not have any discontinuities. Also, f (x) may have as many as four discontinuities. These continuities will only be removable where p and q have roots in common.
(B) Since polynomials of odd degree will have at least one root, f (x) will have at least one discontinuity, and as many as five. But the polynomial need not have any nonremovable discontinuities due to multiplicity of roots.
(C) This situation is the same as the first. Roots of the numerator do not produce discontinuities.