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?
We can factor the given polynomial as follows:
23x^2 + kx - 5 = (23x - 5)(x + 1)
The only way this can be factored into a product of linear binomials with integer coefficients is if k is a multiple of 5. This is because the coefficient of the x term must be divisible by 23, and the coefficient of the constant term must be divisible by 5. The only multiples of 5 that are also multiples of 23 are 5, 10, 15, and 20. Therefore, the possible values of k are 5, 10, 15, and 20.
Here is a table showing the factorization of 23x^2 + kx - 5 for each possible value of k:
k | Factorization
5 | (23x - 5)(x + 1)
10 | (23x - 10)(x + 1)
15 | (23x - 15)(x + 1)
20 | (23x - 20)(x + 1)
