Answer:
Option (b) is correct.
The expression is equivalent, but the \(a^2-4\) term is not completely factored.
Step-by-step explanation:
Given : a student factors \(a^6-64\) to \((a^2-4)(a^4+4a^2+16).\)
We have to choose the correct statement about \((a^2-4)(a^4+4a^2+16).\) from the given options.
Given \(a^6-64\) is factored to \((a^2-4)(a^4+4a^2+16).\)
Consider \(a^6-64\)
Using algebraic identity, \(a^3-b^3=(a-b)(a^2+ab+b^2)\)
comparing \(a=a^2\) and b = 4, we have,
\((a^2)^3-4^3=(a^2-4)(a^4+4a^2+16)\)
Thus, the factorization is equivalent but we can simplify it further also, as
Using algebraic identity, \(a^2-b^2=(a+b)(a-b)\)
Thus, \(a^2-4=a^2-2^2\)
Can be written as \(a^2-4=(a+2)(a-2)\)
Thus, the expression is equivalent, but the \(a^2-4\) term is not completely factored.
Option (b) is correct.
