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)

Question

GPT-4o mini · Answer

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) $**.