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When we simplify a rational expression, the first step is to identify the common factors in both the numerator and the denominator. This makes it easier to reduce the expression to its lowest terms.

A common factor is a number or variable that divides exactly into two or more numbers or terms without leaving a remainder.

For example, in the expression \(\frac{5 y^{2}(y+8)}{15 y(y-8)}\), both 5y and y are common factors in the numerator and denominator.

By identifying these, you set the stage for simplifying the expression further.

To spot common factors, look for numbers that divide both the coefficients and the variables that appear in both the numerator and the denominator.

Factoring a polynomial involves breaking it down into simpler 'factor' polynomials that, when multiplied together, yield the original polynomial.

For the expression \(\frac{5 y^{2}(y+8)}{15 y(y-8)}\), you first factor out the coefficients and the common variables in both the numerator and the denominator.

• The numerator 5y²(y + 8) involves factoring out 5y².
• The denominator 15y(y - 8) involves factoring out 15y.

Factoring helps in identifying parts of the expression that can cancel each other out to simplify the expression.

Simplifying a rational expression often requires finding the Greatest Common Factor (GCF) of the terms involved. The GCF is the largest factor that two or more terms have in common.

In the case of \(\frac{5 y^{2}(y+8)}{15 y(y-8)}\), the GCF is 5y.

Here's how you can identify it:

• First, look at the coefficients: The GCF of 5 and 15 is 5.
• Next, look at the variables: The GCF of y² and y is y.

Combining these, the overall GCF of 5y² and 15y is 5y.

Dividing both the numerator and the denominator by this GCF, the complex expression simplifies to \(\frac{y(y+8)}{3(y-8)}\).

This makes the expression easier to understand and work with.