1. The values are a = 13, b = 13, and c = 13.
2. Yes, the solution to the P(x) problem is unique.
1. Find a, b, and c.
Since P(x) is divisible by (x - c), this means that P(c) = 0. Therefore, we can substitute c into the expression for P(x) and set it equal to zero:
c(c - a)(c - b) - 13 = 0
Expanding this expression, we get:
c^3 - (a + b)c^2 + (ab - 13)c = 0
We can factor this expression as:
(c - 13)(c^2 - (a + b)c + ab) = 0
This gives us two factors:
c - 13 = 0 or c^2 - (a + b)c + ab = 0
Solving the first factor, we get c = 13.
Substituting c = 13 into the second factor, we get:
13^2 - (a + b)13 + ab = 0
Expanding this expression, we get:
169 - 13(a + b) + ab = 0
This simplifies to:
ab - 13a - 13b + 169 = 0
We can rewrite this expression as:
ab - 13a - 13b + 169 - 13 = 0
Factoring out a - 13, we get:
(a - 13)(b - 13) = 0
This gives us two factors:
a - 13 = 0 or b - 13 = 0
Solving the first factor, we get a = 13.
Solving the second factor, we get b = 13.
Therefore, a = 13, b = 13, and c = 13.
2. Is your solution unique? Justify your answer.
Yes, the solution to the P(x) problem is unique. This is because there is only one set of values for a, b, and c that satisfies the given conditions.
To see why, we can rewrite the given conditions as a system of three equations:
0 < a < b
P(c) = 0
We have already found that the values a = 13, b = 13, and c = 13 satisfy these conditions. To show that these values are unique, we can prove that no other set of values satisfies the conditions.
Consider the inequality 0 < a < b. This inequality implies that a and b cannot be equal. Additionally, since a and b are integers, they cannot both be negative. Therefore, a and b must be distinct positive integers.
Now consider the equation P(c) = 0. Substituting P(x) = x(x - a)(x - b) - 13 into this equation, we get:
c(c - a)(c - b) - 13 = 0
Since c is an integer, this equation implies that (c - 13) is a factor of (c^2 - (a + b)c + ab). This means that (c - 13) must divide both c^2 - (a + b)c and ab.
Since c is an integer, (c - 13) cannot be a fraction. Therefore, (c - 13) must divide both c^2 - (a + b)c and ab. This means that (c - 13) must be a factor of a and b.
However, we have already seen that a and b must be distinct positive integers. Therefore, (c - 13) cannot be a factor of both a and b. The only way this can be true is if (c - 13) is equal to 1. In other words, c = 14.
However, this contradicts the condition that 0 < a < b, since a = 13. Therefore, there are no other values for a, b, and c that satisfy the given conditions.
In conclusion, the solution to the P(x) problem is unique. The only set of values that satisfies the given conditions is a = 13, b = 13, and c = 13.
The following question may be like this:
Exploration 2.2.3: The P(x) Problem [A concept connections chal- lenge] Given integers a, b, and c such that 0<a<b, and given P(x) = x(x - a)(x - b) - 13 where P(x) is divisible by (x - c). 1. Find a, b, and c. 2. Is your solution unique? Justify your answer.