If x^7 - x^3 = 1382, how many values are possible for x?

Steps to Solve

1. Rearranging the Equation
   We start with the equation:
   $$ x^7 - x^3 = 1382 $$
   Rearranging gives:
   $$ x^7 - x^3 - 1382 = 0 $$

2. Factoring Out x^3
   Next, we can factor out $x^3$ from the first two terms:
   $$ x^3(x^4 - 1) - 1382 = 0 $$
   This can be further simplified as:
   $$ x^3(x^2 - 1)(x^2 + 1) - 1382 = 0 $$

3. Finding Possible Integer Solutions
   To find integer solutions, we need to check values of $x$ that can satisfy $x^7 - x^3 = 1382$. We will test integer values around the 7th root of 1382.
   First, calculate the approximate value:
   $$ x \approx 5.29 $$
   So we will check integer values $x = 4, 5, 6$.

4. Testing integer values

• For $x = 4$:
  $$ 4^7 - 4^3 = 16384 - 64 = 16320 \quad (\text{not } 1382) $$

• For $x = 5$:
  $$ 5^7 - 5^3 = 78125 - 125 = 78000 \quad (\text{not } 1382) $$

• For $x = 6$:
  $$ 6^7 - 6^3 = 279936 - 216 = 279720 \quad (\text{not } 1382) $$

• For $x = 3$:
  $$ 3^7 - 3^3 = 2187 - 27 = 2160 \quad (\text{not } 1382) $$

• For $x = 2$:
  $$ 2^7 - 2^3 = 128 - 8 = 120 \quad (\text{not } 1382) $$

After testing these values, note that none of these values satisfy the equation.

5. Conclusion on Possible Values
   Since no integer solution satisfies the equation, the number of possible values for $x$ is $0$.