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The correct answer for the given question above with the given polynomial is the first option: Difference of Cubes. Difference of cubes is the polynomial identity that should be used to prove that 117 = 125 - 8. Difference of cubes is the difference of a perfect cube and another perfect cube.

Answer:

The polynomial identity that should be used to prove :117 = 125-8 is;

Difference of cubes

Explanation:

Polynomial identities are just equations that are true, but identities are particularly useful for showing the relationship between two apparently independent expressions.

Difference of cube:

\(a^3-b^3=(a-b)(a^2+ab+b^2)\).

To prove: 117 =125 -8

Take RHS;

125 -8, we can write this as:

\(5^3 -2^3\)  

Now, using polynomial identity of difference of cubes in above expression:

125-8 =\(5^3 -2^3 =(5-2)(5^2+5\cdot 2+2^2)\) = \(3 \cdot (25+10+4) = 3 \cdot 39 =117\)

RHS =LHS                  Hence Proved!



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