Question

We are looking to factor the quadratic $23x^2 + kx - 5.$ Some values of $k$ allow us to factor it as a product of linear binomials with integer coefficients. What are all such values of $k?$

Answer 1

We need (ax + b)(cx + d) = 23x^2 + kx - 5 with integers a,c,b,d. Then ac = 23 and bd = -5. Since 23 is prime, (a,c) = (1,23) or (23,1) (up to sign), and bd ∈ {1·(−5), (−1)·5, 5·(−1), (−5)·1}. Computing k = ad + bc for each choice yields

- b = 1, d = −5: k = 23·1 + (−5) = 18, factors (x+1)(23x−5).
- b = −1, d = 5: k = −23 + 5 = −18, factors (x−1)(23x+5).
- b = 5, d = −1: k = 115 − 1 = 114, factors (x+5)(23x−1).
- b = −5, d = 1: k = 1 − 115 = −114, factors (x−5)(23x+1).

Thus all integer k for which the quadratic factors over Z are k = 18, −18, 114, −114.