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When simplifying rational expressions, the first step is to look for common factors in the numerator and the denominator. Common factors are terms that appear in both parts of the fraction. Identifying these factors allows us to simplify the expression more easily. In the exercise, the rational expression is \(\frac{(2x+7)(x-1)}{(2x+3)(2x+7)}\). Here, \((2x+7)\) is a common factor because it appears both in the numerator and the denominator. Spotting these common factors helps streamline the process of simplifying rational expressions. Watch out for these common factors in any rational expression. They will help you reduce the expression to its lowest terms effectively.

Once you identify common factors in both the numerator and the denominator, the next step is to cancel them. Canceling is just removing the common factors from both parts of the fraction. This does not change the value of the expression, but it simplifies it.

In our example rational expression, \(\frac{(2x+7)(x-1)}{(2x+3)(2x+7)}\), the common factor is \((2x+7)\). So, we can cancel \((2x+7)\) from both the top and the bottom.

After canceling, the simplified expression is \(\frac{x-1}{2x+3}\). Canceling common factors is a crucial step to simplify rational expressions and should always be done carefully to avoid any mistakes. Remember, only terms that are factors, meaning they multiply together to form the numerator or denominator, can be canceled.

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying these expressions involves reducing them to their lowest terms, much like simplifying numerical fractions.

Let's break down the provided example further: \(\frac{(2x+7)(x-1)}{(2x+3)(2x+7)}\). On first glance, it might seem complicated, but recognizing common factors is key.

Step-by-step:

• First, identify any common factors (like we did with \((2x+7)\)).
• Then, cancel these factors to reduce the expression.

After the common factors \((2x+7)\) are canceled, the expression reduces to \(\frac{x-1}{2x+3}\). Rational expressions often look complex at first sight, but by identifying and canceling common factors, they can be greatly simplified. This makes them easier to handle and solve in further mathematical problems.