Final answer:
The possible values of k for x² + kx + 18 to be factorable are 19, -19, 11, -11, 9, and -9, which correspond to the sums of the pairs of factors of 18.
Step-by-step explanation:
If the quadratic expression x² + kx + 18 is factorable over the integers, the possible values of k should be such that the constant term 18 can be expressed as the product of two integers whose sum is equal to the value of k. The constant term, 18, has the following pairs of factors: (1, 18), (-1, -18), (2, 9), (-2, -9), (3, 6), and (-3, -6). For x² + kx + 18 to factor into a product of two binomial terms, (x + m)(x + n), where m and n are factors of 18, the sum m + n must equal k.
To find the possible values for k, we consider the sums of all pairs of factors:
- 1 + 18 = 19
- -1 - 18 = -19
- 2 + 9 = 11
- -2 - 9 = -11
- 3 + 6 = 9
- -3 - 6 = -9
Thus, the possible values of k for the given quadratic expression to be factorable are 19, -19, 11, -11, 9, and -9.