Introduction
To find the value of 'm' in the given polynomial equation (m^2 4)x^2 65x 4m, where one of the zeros is the reciprocal of the other.

Solution
Let the two zeros of the given polynomial be a and 1/a.

Sum of the roots
The sum of the roots of the quadratic equation is given by -b/a, where b is the coefficient of x and a is the coefficient of x^2.

Therefore, a + (1/a) = -65/(m^2 4)

Product of the roots
The product of the roots of the quadratic equation is given by c/a, where c is the constant term and a is the coefficient of x^2.

Therefore, a(1/a) = 4m/(m^2 4) = 4/(m)

Since one root is the reciprocal of the other, we have a = 1/a.

Therefore, a^2 = 1, which implies that a = 1 or a = -1.

Case 1: a = 1
If a = 1, then the other root is also 1, which is not possible since we know that one root is the reciprocal of the other.

Case 2: a = -1
If a = -1, then the other root is -1, which satisfies the condition that one root is the reciprocal of the other.

Substituting a = -1 in the equation a + (1/a) = -65/(m^2 4), we get:

-1 + (-1) = -65/(m^2 4)

-2 = -65/(m^2 4)

m^2 4 = 65/2

m^2 = 33/2

m = ±√(33/2)

Therefore, the value of 'm' is ±√(33/2).

Conclusion
Thus, we have found the value of 'm' in the given polynomial equation (m^2 4)x^2 65x 4m, where one of the zeros is the reciprocal of the other.