The expression \((a - 2)^3\) does not equal \(a^3 - 8\). Let's break it down to understand why this is incorrect.
When we expand \((a - 2)^3\), we can use the binomial theorem or simply apply the formula for the cube of a binomial:
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
In this case, \(a\) is \(a\) and \(b\) is \(2\):
\[ (a - 2)^3 = a^3 - 3(a)(2) + 3(2^2)(a) - 2^3 \] \[ = a^3 - 3 \cdot 2a^2 + 3 \cdot 4a - 8 \] \[ = a^3 - 6a^2 + 12a - 8 \]
So the correct expansion of \((a - 2)^3\) is:
\[ (a - 2)^3 = a^3 - 6a^2 + 12a - 8 \]
The expression \(a^3 - 8\) is actually the result of the difference of cubes, which can be factored as:
\[ a^3 - 2^3 = (a - 2)(a^2 + 2a + 4) \]
Thus, it's clear that \((a - 2)^3\) results in a polynomial that includes more terms than just \(a^3 - 8\). Therefore, the assertion \((a - 2)^3 = a^3 - 8\) is incorrect.
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