Verified Answer :

  • The expression \(27v^6 - 8x^6\) is a difference of cubes: \((3v^2)^3 - (2x^2)^3\).
  • Using the difference of cubes formula \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) with \(a=3v^2\) and \(b=2x^2\), we factor it into \((3v^2 - 2x^2)(9v^4 + 6v^2x^2 + 4x^4)\).
  • Final factored form: \(\boxed{(3v^2 - 2x^2)(9v^4 + 6v^2x^2 + 4x^4)}\).

Explanation

  1. Identifying the Structure of the Expression

To factor the expression \(27v^6 - 8x^6\), we first observe its structure. The expression is a difference of two terms, both of which are perfect cubes.
We can write the terms as follows:
\[27v^6 = (3v^2)^3\]
\[8x^6 = (2x^2)^3\]
Thus, the expression is of the form \(a^3 - b^3\), where \(a = 3v^2\) and \(b = 2x^2\).

  1. Applying the Difference of Cubes Factorization

The difference of cubes formula states that:
\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
Substituting \(a = 3v^2\) and \(b = 2x^2\) into the formula, we have:
\[(3v^2)^3 - (2x^2)^3 = (3v^2 - 2x^2)((3v^2)^2 + (3v^2)(2x^2) + (2x^2)^2)\]
Now, we simplify the terms inside the second factor:
\((3v^2)^2 = 9v^4\)
\((3v^2)(2x^2) = 6v^2x^2\)
\((2x^2)^2 = 4x^4\)
Combining these, we get:
\[(3v^2 - 2x^2)(9v^4 + 6v^2x^2 + 4x^4)\]
This is the factored form of the original expression.

Examples

Factoring expressions like \(27v^6 - 8x^6\) is essential in engineering and physics, where complex equations are often simplified to solve for variables. For instance, in structural mechanics, analyzing the stresses and strains on beams or plates often involves polynomial expressions that must be factored to find critical points or solve for dimensions. By reducing these expressions into simpler factors, engineers can easily determine safe design limits, optimize material usage, or identify physical phenomena like resonance in a system. Mastery of algebraic techniques ensures that these calculations are performed accurately and efficiently.

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