Final answer:
To determine when x² + kx + 12 is not prime, one must find pairs of integers (p, q) that satisfy p + q = k and p × q = 12. The possibilities for k are 13, 8, 7, -13, -8, and -7.
Step-by-step explanation:
The student is asking for all possibilities for the constant k such that the quadratic expression x² + kx + 12 is not prime. A quadratic expression is prime if it cannot be factored into a product of two linear expressions with integer coefficients. The expression given can be factored if k is such that there exist integers p and q where p + q = k and p × q = 12. The possible pairs (p, q) that satisfy these conditions are (1, 12), (2, 6), (3, 4), (-1, -12), (-2, -6), and (-3, -4). Therefore, the possible values for k are 1 + 12, 2 + 6, 3 + 4, -1 - 12, -2 - 6, and -3 - 4, which simplifies to k being 13, 8, 7, -13, -8, and -7 respectively.
To determine the values of k for which the quadratic expression x² + kx + 12 is not prime, we need to find the values of k that result in the expression not being a prime number for any value of x.
If we factorize the expression, we get (x + 3)(x + 4). Therefore, the expression is not prime when k = 3 or k = 4, as these values will result in the quadratic having factors other than 1 and itself.
For example, when k = 3, the expression becomes x² + 3x + 12, which factorizes to (x + 3)(x + 4). When k = 4, the expression becomes x² + 4x + 12, which also factorizes to (x + 3)(x + 4).