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Factoring is an essential technique when working with polynomials and rational expressions. It involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. It’s like reverse multiplying — you are looking for numbers or expressions that multiply to give what you started with.

When dealing with rational expressions, factoring helps simplify them. In our exercise, the numerator is already factored as \(r - 4\). The denominator, however, \(r^2 - 16\), needs to be factored. By identifying it as a special case of the difference of squares, we can easily write it as \((r + 4)(r - 4)\).

• Look for common factors first.
• Identify special cases like the difference of squares or perfect square trinomials.

By factoring, we can often cancel out terms, which leads us to simplifying expressions or finding a way to write them in their simplest form.

The difference of squares is a special factoring pattern you can use when you see an expression in the form of \(a^2 - b^2\). This pattern breaks down into two binomials: \((a + b)(a - b)\). This is a quick and efficient way to simplify expressions.

In our example, the expression \(r^2 - 16\) fits this pattern perfectly because \(r^2\) is \(r\times r\) and 16 is a perfect square \((4\times 4)\). This allows us to factor the expression into \((r + 4)(r - 4)\).

• Identify the expression as a difference of squares.
• Apply the pattern \(a^2 - b^2 = (a + b)(a - b)\) for quick factoring.

By recognizing these patterns, you can simplify expressions quickly, making it easier to work with complex rational expressions.

Simplifying a rational expression to its lowest terms means reducing it so that the numerator and denominator have no common factors other than 1, essentially making the expression simpler without changing its value.

In the example given, the expression \(\frac{r-4}{(r+4)(r-4)}\) has a common factor in both the numerator and the denominator, which is \(r-4\). By canceling this out, you are left with \(\frac{1}{r+4}\), which is the expression in its simplest or lowest terms.

• Factor both the numerator and the denominator.
• Cancel out any common factors between them.
• Check to make sure there are no more common factors.

When you reduce expressions to their lowest terms, it simplifies the expression and helps you understand its fundamental components better.