To determine which of the provided expressions is the general formula for factoring the difference of cubes, we recall that the difference of cubes can be written as [tex]\( a^3 - b^3 \)[/tex].
The given formula options are:
1. [tex]\( (a-b) \left(a^2 + ab + b^2 \right) \)[/tex]
2. [tex]\( (a+b) \left(a^2 - ab + b^2 \right) \)[/tex]
3. [tex]\( (a+b) \left(a^2 + ab + b^2 \right) \)[/tex]
4. [tex]\( (a-b) \left(a^2 - ab - b^2 \right) \)[/tex]
Let's recall the general factoring formulas for sum and difference of cubes:
- For [tex]\( a^3 - b^3 \)[/tex] (difference of cubes):
[tex]\[
a^3 - b^3 = (a - b) \left(a^2 + ab + b^2 \right)
\][/tex]
- For [tex]\( a^3 + b^3 \)[/tex] (sum of cubes):
[tex]\[
a^3 + b^3 = (a + b) \left(a^2 - ab + b^2 \right)
\][/tex]
Based on these formulas, we see that the correct formula for factoring the difference of cubes [tex]\( a^3 - b^3 \)[/tex] is:
[tex]\[
(a - b) \left(a^2 + ab + b^2 \right)
\][/tex]
Among the provided options, the correct one that matches this formula is:
[tex]\[
(a - b) \left(a^2 + ab + b^2 \right)
\][/tex]
Therefore, the correct answer is:
1. [tex]\( (a-b) \left(a^2 + ab + b^2 \right) \)[/tex]
