To find the lowest common multiple (LCM) of the expressions [tex]\(x - 4\)[/tex] and [tex]\((x-4)(x+4)\)[/tex], let's carefully go through the steps.
1. Identifying the expressions:
- The first expression is [tex]\(x - 4\)[/tex].
- The second expression is [tex]\((x - 4)(x + 4)\)[/tex].
2. Factoring the expressions:
- The first expression [tex]\(x - 4\)[/tex] is already factored.
- The second expression [tex]\((x - 4)(x + 4)\)[/tex] is a product of two binomials.
3. Understanding the relationship between the expressions:
- Notice that [tex]\((x - 4)(x + 4)\)[/tex] is simply the expansion of the difference of squares formula, which gives [tex]\(x^2 - 16\)[/tex].
4. Writing the expressions in their factored forms:
- The first expression remains [tex]\(x - 4\)[/tex].
- The second expression can be written as [tex]\(x^2 - 16\)[/tex], which is already in a product form, and is fundamentally equivalent to the product of [tex]\((x - 4)\)[/tex] and [tex]\((x + 4)\)[/tex].
5. Determining the LCM:
- The LCM of two expressions is the smallest expression (or polynomial) that both original polynomials divide into without leaving a remainder.
- [tex]\(x - 4\)[/tex] is a factor of both [tex]\(x - 4\)[/tex] and [tex]\(x^2 - 16\)[/tex].
- [tex]\(x^2 - 16\)[/tex] (which is expanded from [tex]\((x - 4)(x + 4)\)[/tex]) naturally includes [tex]\(x - 4\)[/tex] as a factor.
6. Conclusion:
- The simplest expression that includes both factors [tex]\(x - 4\)[/tex] and [tex]\((x - 4)(x + 4)\)[/tex] without leaving any remainders is [tex]\(x^2 - 16\)[/tex].
Therefore, the lowest common multiple of [tex]\(x - 4\)[/tex] and [tex]\((x - 4)(x + 4)\)[/tex] is:
[tex]\[ \boxed{x^2 - 16} \][/tex]
