Question

Divide the polynomial x3−a3 by x−a . Which polynomial identity does this establish?(1 point)
Responses

x3−a3=(x−a)(x2+ax−a2)
x cubed minus a cubed equals left parenthesis x minus a right parenthesis left parenthesis x squared plus a x minus a squared right parenthesis

x3−a3=(x−a)(x2+ax+a2)
x cubed minus a cubed equals left parenthesis x minus a right parenthesis left parenthesis x squared plus a x plus a squared right parenthesis

x3−a3=(x−a)(x2−ax−a2)
x cubed minus a cubed equals left parenthesis x minus a right parenthesis left parenthesis x squared minus a x minus a squared right parenthesis

x3−a3=(x−a)(x2−ax+a2)
x cubed minus a cubed equals left parenthesis x minus a right parenthesis left parenthesis x squared minus a x plus a squared right parenthesis

Answer 1

To divide the polynomial \(x^3 - a^3\) by \(x - a\), we can use polynomial long division or apply the fact that \(x^3 - a^3\) is a difference of cubes.

The difference of cubes can be factored using the formula:

\[
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
\]

This indicates that when you divide \(x^3 - a^3\) by \(x - a\), the result is the polynomial \(x^2 + ax + a^2\).

Thus, the correct identity is:

\[
x^3 - a^3 = (x - a)(x^2 + ax + a^2)
\]

This identity establishes the relationship between the difference of cubes and its factorization.

So the correct answer from the options provided is:

**\(x^3 - a^3 = (x - a)(x^2 + ax + a^2)\)**