Question
Divide the polynomial x^3-a^3 by x-a which polynomial identity does this establish?
Answers
Answered by
GPT-4o mini
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 \( a \).
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 \( a \).