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To simplify rational expressions, the first step is identifying common factors in the numerator and the denominator. A factor is a number or expression that divides another number or expression evenly. In the given problem, the numerator is \((t+5)(t-3)\) and the denominator is \((t+5)^2\). Here, both the numerator and denominator share a common factor of \((t+5)\). Being able to recognize these common factors is crucial for simplifying rational expressions.

When examining the expressions, look for common terms or polynomial expressions that can be factored out. It helps to write out all the factors explicitly to see the common ones.

In future exercises, practice identifying common factors by:

• Breaking down each term into its prime factors or simpler components.
• Highlighting or underlining commonalities.

Identifying common factors sets the stage for canceling, which is the next important step.

Once you have identified the common factors, the next step is to cancel them out. In our example, we found that both the numerator \((t+5)(t-3)\) and the denominator \((t+5)^2\) share a common factor of \((t+5)\). Canceling this common factor simplifies the rational expression.

To cancel a common factor:

• Strike through the common factor in both the numerator and the denominator.
• Rewrite the remaining expression.

For this problem, canceling \((t+5)\) from both the numerator and the denominator gives us \( \frac{(t-3)}{(t+5)} \).

Make sure you only cancel common factors. Both the numerator and denominator should be divisible by the same factor. If not evenly divisible, re-evaluate the factorization.

Practice canceling common factors by:

• Using simple numbers first, and gradually moving to more complex algebraic expressions.
• Double-checking your work to ensure no mistakes are made in canceling.

A rational expression is simply a fraction where both the numerator and the denominator are polynomials. Simplifying rational expressions involves several steps: finding common factors, canceling them, and then rewriting the expression.

To put it all together using our example: We started with \( \frac{(t+5)(t-3)}{(t+5)^2} \). First, we found the common factor \((t+5)\). Then, we canceled this factor from both parts of the fraction. Finally, the simplified rational expression is \( \frac{t-3}{t+5} \).

Simplifying rational expressions is useful for solving equations and can make working with algebraic fractions much easier. Always follow these general steps:

• Identify common factors.
• Cancel the common factors.
• Rewrite the simplified expression.

Remember, taking these steps helps ensure your results are as simplified and accurate as possible.