To divide the polynomial \( x^3 - a^3 \) by \( x - a \), we can use polynomial long division or apply the factorization of the difference of cubes.
The difference of cubes can be factored as:
\[ x^3 - a^3 = (x - a)(x^2 + ax + a^2) \]
This identity shows that \( x^3 - a^3 \) can be expressed as the product of \( x - a \) and \( x^2 + ax + a^2 \).
Now, to confirm the division:
\[ \frac{x^3 - a^3}{x - a} = x^2 + ax + a^2 \]
Thus, the polynomial identity that this establishes is:
\[ x^3 - a^3 = (x - a)(x^2 + ax + a^2) \]
This identity applies for any real or complex number \( a \).
1 answer