Answer:
Plus or minus: 13, 12, 11, 10, 9, 8, 7, 3, 2
Step-by-step explanation:
We need to find two integers, let's say a and b, such that:
3x^2 + px + 4 = (ax + c)(bx + d)
Expanding the right-hand side, we get:
3x^2 + px + 4 = abx^2 + (ad + bc)x + cd
So we need to find integers a, b, c, and d that satisfy:
ab = 3
ad + bc = p
cd = 4
The first equation tells us that either a = 1, b = 3 or a = 3, b = 1 (since 3 is prime, its only factors are 1 and 3, up to sign). Let's consider both cases:
Case 1: a = 1, b = 3
Then ab = 3 implies that a = 1 and b = 3, or a = -1 and b = -3. The third equation (cd = 4) then implies that c and d are either (1, 4) or (-1, -4), up to sign. Substituting these values into the second equation (ad + bc = p), we get:
1d + 3c = p or -1d - 3c = p
We can simplify these equations by multiplying the second one by -1:
-1d - 3c = p
1d + 3c = -p
Adding the two equations, we get:
0 = 0
This means that the value of p does not matter in this case, and we can always find integers a, b, c, and d that satisfy the equation.
Case 2: a = 3, b = 1
Then ab = 3 implies that a = 3 and b = 1, or a = -3 and b = -1. The third equation (cd = 4) then implies that c and d are either (1, 4) or (-1, -4), up to sign. Substituting these values into the second equation (ad + bc = p), we get:
3d + 1c = p or -3d - 1c = p
We can simplify these equations by multiplying the second one by -1:
-3d - 1c = p
3d + 1c = -p
Adding the two equations, we get:
0 = 0
This means that the value of p does not matter in this case either.
Therefore, we can always find integers a, b, c, and d that satisfy the equation, for any value of p.
Answer: Plus or minus: 13, 12, 11, 10, 9, 8, 7, 3, 2
