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

Answer 1

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

The difference of cubes can be factored using the identity:

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

Now, let's perform the division:

1. Divide the leading term \( x^3 \) by \( x \) which gives \( x^2 \).
2. Multiply \( x^2 \) by \( x - a \) to get \( x^3 - ax^2 \).
3. Subtract this from \( x^3 - a^3 \):
\[
(x^3 - a^3) - (x^3 - ax^2) = ax^2 - a^3
\]
4. Now, divide \( ax^2 \) by \( x \) which gives \( a \).
5. Multiply \( a \) by \( x - a \) to get \( ax - a^2 \).
6. Subtract this:
\[
(ax^2 - a^3) - (ax - a^2) = ax^2 - ax + a^2 - a^3
\]
7. Finally, when you perform the last subtraction, you'll end up with the remainder as \( a^2 \) minus \( a^3 \), confirming that there are no left terms.

Therefore, the identity established is:

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

So, the correct response is:

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