- Analyze the given expression \(\(x^6 - 9\)\) and compare it with the provided options.
- Determine that option \(\(b\)\), \(\((x^3)^2 - 3^2\)\), is the correct match.
- Recognize the form \(\(A^2 - B^2\)\) as a standard 'difference of squares'.
- Conclude that the expression is a \(\boxed{\text{difference of squares}}\).
Explanation
- Analyzing the Expression
We are given the polynomial expression \(x^6 - 9\). Our goal is to determine which of the provided options is equivalent to this expression and identify the type of polynomial it represents. The expression is given as a binomial, and we can look for ways to rewrite its terms as squares or cubes to identify a familiar algebraic form.
- Evaluating the Options
We evaluate each option by simplifying it:
- Option (a): \((x^3)^2 - 3^3 = x^6 - 27\). This is not equivalent to \(x^6 - 9\).
- Option (b): \((x^3)^2 - 3^2 = x^6 - 9\). This is equivalent to our given expression.
- Option (c): \((x^3)^3 - 3^2 = x^9 - 9\). This is also not equivalent to \(x^6 - 9\).
Comparing these with the target expression, we see that option (b) is the correct choice.
- Classifying the Polynomial
The expression \((x^3)^2 - 3^2\) is in the form \(A^2 - B^2\), where \(A = x^3\) and \(B = 3\). In algebra, an expression in the form \(A^2 - B^2\) is known as a 'difference of squares'. This type of polynomial can be factored as \((A - B)(A + B)\), or in this case, \((x^3 - 3)(x^3 + 3)\). Therefore, the polynomial is a difference of squares.
- Conclusion
The equivalent expression is \((x^3)^2 - 3^2\) (Option b), and the polynomial is a difference of squares. Thus, the final answer for the blank is \(\boxed{\text{difference of squares}}\).
Examples
Identifying the structure of polynomials like the difference of squares is a fundamental skill in algebra, commonly used to simplify complex equations or solve quadratic equations more efficiently. For instance, if you are designing a physical component and need to calculate the difference in area between two square-shaped surfaces with side lengths \(x^3\) and 3, you would use this algebraic identity to quickly determine the result: \((x^3)^2 - 3^2 = (x^3 - 3)(x^3 + 3)\). This technique is also vital in higher-level mathematics for integration and solving various physics problems.