Find the sum of all integers c, such that for some integers a and b satisfying a<b<c,
a(b−c)^4+b(c−a)^4+c(a−b)^4=836
cheating
what is the answer correct?
To find the sum of all integers c that satisfy the given equation, we need to find the values of c for which the equation holds true.
Let's begin by expanding the equation:
a(b - c)^4 + b(c - a)^4 + c(a - b)^4 = 836
Next, we can simplify the equation by expanding and rearranging the terms:
a(b^4 - 4b^3c + 6b^2c^2 - 4bc^3 + c^4) + b(c^4 - 4c^3a + 6c^2a^2 - 4ca^3 + a^4) + c(a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4) = 836
By rearranging and combining like terms, the equation becomes:
(-3a^3 - 3b^3 - 3c^3) + (6a^2b + 6ab^2) + (-4a^3 - 4b^3 + 6a^2c + 6b^2c + 12abc) + (6ab^2 + 6b^2c) + (-4a^3 - 4b^3 + 6ac^2 + 6bc^2 + 12abc) + (-3a^3 - 3b^3 - 3c^3) = 836
Combining like terms further, we get:
-14a^3 + 12a^2b + 12ab^2 + 6a^2c + 18abc + 6b^2c + 6ac^2 + 6bc^2 - 14b^3 + 12ab^2 - 14c^3 + 6bc^2 = 836
Simplifying:
-14a^3 + 12a^2b + 18abc + 6a^2c + 6ac^2 + 12ab^2 - 14b^3 + 12ab^2 - 14c^3 + 12b^2c = 836
Now, we observe that the left-hand side of the equation is symmetric with respect to the variables a, b, and c. This suggests that if a, b, and c satisfy the equation, then any permutation of a, b, and c will also satisfy the equation. Therefore, we can assume without loss of generality that a ≤ b ≤ c.
Since a, b, and c are integers satisfying the inequality a < b < c, we can start by considering the lowest possible value for a.
Suppose a = 1:
Plugging in a = 1, the equation becomes:
-14(1)^3 + 12(1)^2b + 18(1)b + 6(1)^2c + 6(1)c^2 + 12(1)b^2 - 14b^3 - 14c^3 + 12b^2c = 836
Simplifying:
-14 + 12b + 18b + 6c + 6c^2 + 12b^2 - 14b^3 - 14c^3 + 12b^2c = 836
Rearranging and combining like terms:
-14c^3 + 6c^2 + 12b^2c + 836 = 14b^3 - 30b - 30
The above equation is a cubic polynomial equation in terms of c. By solving this equation, we can find the values of c that satisfy it. Since the equation includes a cubic term, we may find multiple values of c for a certain value of b.
We can repeat this process for each value of a starting from 1. In each case, we would solve the resulting cubic equation to find the values of c that satisfy it. After obtaining the sets of solutions for each a, we can sum up all the distinct values of c to get the final answer.
Please note that solving cubic polynomial equations can be a complex process and may require further mathematical techniques, such as factoring, synthetic division, or using software or computer algebra systems.