For each of the following expressions, determine all of the values of k that allow the trinomial to be factored over the integers.
A) x^2 + kx - 19
B) 25x^2 + kx + 49
C) x^2 + kx + 8
(x-19)(x+1)....k=18
(5x+7)^2 k=70
(25x+7)(x+7) k=181, or
(25x-7)(x-7) k=-181
(5x+49)(5x+1) k=150 or -150
(25x+49)(x+1) k=74 or -74
(25x+1)(x+49) k=1275 or -1275
To determine all the values of k that allow each trinomial to be factored over the integers, we need to check if there are any integer roots for each equation. We can use the factoring technique or the quadratic formula to solve for x and find the values of k.
A) x^2 + kx - 19
To factor this trinomial, we need two integers that multiply to -19 and add up to k. The pairs of integers whose product is -19 are (1, -19) and (-1, 19). The pairs that add up to k are (20, -19) and (-20, 19). So, the possible values of k are 20 and -20.
B) 25x^2 + kx + 49
To factor this trinomial, we need two integers that multiply to 49*25 = 1225 and add up to k. The pairs of integers whose product is 1225 are (1, 1225), (-1, -1225), (5, 245), (-5, -245), (7, 175), (-7, -175), (25, 49), and (-25, -49). The pairs that add up to k are (1226, 1), (-1226, -1), (250, -245), (-250, 245), (182, -175), (-182, 175), (74, -49), and (-74, 49). So, the possible values of k are 1226, -1226, 250, -250, 182, -182, 74, and -74.
C) x^2 + kx + 8
To factor this trinomial, we need two integers that multiply to 8 and add up to k. The pairs of integers whose product is 8 are (1, 8), (-1, -8), (2, 4), and (-2, -4). The pairs that add up to k are (9, 1), (-9, -1), (6, 2), and (-6, -2). So, the possible values of k are 9, -9, 6, and -6.
Therefore, the values of k that allow each trinomial to be factored over the integers are:
A) k = 20, -20
B) k = 1226, -1226, 250, -250, 182, -182, 74, -74
C) k = 9, -9, 6, -6