Question

Prove that \(x^n - y^n\) is divisible by \(x+y\) only when \(n\) is even.

Hint:

To check whether a polynomial is divisible by \(x+a\), substitute \(x=-a\). If the result is zero, then the polynomial is divisible by that factor.

Answer 1

Step 1: Use the divisibility condition.}
If a polynomial is divisible by \(x+y\), then on putting \(x=-y\), the polynomial must become zero.
So, consider:
\[ x^n - y^n \]
Step 2: Substitute \(x=-y\).}
Replacing \(x\) by \(-y\), we get:
\[ (-y)^n - y^n \]
Step 3: Analyze according to whether \(n\) is even or odd.}
Now, \((-y)^n\) depends on the parity of \(n\):
If \(n\) is even, then:
\[ (-y)^n = y^n \] So,
\[ (-y)^n - y^n = y^n - y^n = 0 \] Hence, \(x+y\) divides \(x^n-y^n\).
If \(n\) is odd, then:
\[ (-y)^n = -y^n \] So,
\[ (-y)^n - y^n = -y^n - y^n = -2y^n \neq 0 \] Hence, \(x+y\) does not divide \(x^n-y^n\).

Step 4: State the conclusion.}
Therefore, \(x^n - y^n\) is divisible by \(x+y\) only when \(n\) is even.

Step 5: Final statement.}
Thus, we have proved that:
\[ \boxed{x^n-y^n \text{ is divisible by } x+y \text{ only if } n \text{ is even}} \]