Final answer:
The polynomial p(x) has zeros of 5 and 1/5. The product of the zeros (k/5) equals 1, which implies that k equals 5.
Step-by-step explanation:
If one of the zeros of the polynomial p(x)=5x²−13x+k is equal to 5, and the other is its reciprocal, then the other zero must be 1/5. The product of the zeros of a quadratic equation ax² + bx + c = 0 is c/a. Given that the zeros of this polynomial are 5 and 1/5, their product is 5 × 1/5 = 1. According to the relationship between coefficients and zeros, we have:
Given that one of the zeros of the polynomial is equal to 5 and the other is its reciprocal, we can determine the value of k. Let's assume the other zero is 1/5. Using the sum and product of zeros concept, we know that the sum of zeros is given by (-b)/a and the product of zeros is given by c/a. In this case, the sum of zeros is (-13)/5 and the product of zeros is k/5. Setting up the equations and solving, we get (-13)/5 + 1/5 = k/5 and (-13)/5 * 1/5 = k/5. Simplifying further, we have -2/5 = k/5 and k = -2.
k/5 = 1
Therefore, k equals 5. The value of k that satisfies the given conditions of the polynomial p(x) is 5.