We are looking to factor \(23x^2 + kx - 5\) Some values of \(k\) allow us to factor it as a product of linear binomials with integer coefficients. What are all such values of k.
In order for 23x^2 + kx - 5 to be factorable as a product of linear binomials with integer coefficients, the following conditions must be met:
(1) The coefficient on the x^2 term must be divisible by 2.
(2) The constant term must be divisible by 5.
(3) The coefficient on the x term must be divisible by the product of the other two coefficients.
This means that k must be divisible by 2 and 5, and it must be such that 5 divides 23k - 5. The only values of k that satisfy these conditions are k=−2,−7,−12,−17,−22.