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Factorization is the process of breaking down a mathematical expression into a product of its factors.
The factors are numbers or expressions that when multiplied together give the original expression.
This step is crucial for simplifying rational expressions because it helps us identify common factors that we can cancel out.

In our example, let's take the numerator: \(2x^2 - 5x\).
We look for the greatest common factor (GCF) of the terms in the numerator.
The terms are \(2x^2\) and \(-5x\).
The GCF is the highest factor that divides both \(2x^2\) and \(-5x\) without leaving a remainder.
Here, the GCF is \(x\).
So, we factor out \(x\) to get: \(x(2x - 5)\).

The same process applies to the denominator: \(16x - 40\).We need to find the GCF of the terms \(16x\) and \(-40\).
The GCF here is \(8\).
By factoring out \(8\) from the denominator, we get: \(8(2x - 5)\).Now, the rational expression becomes \(\frac{x(2x-5)}{8(2x-5)}\).

The Greatest Common Factor (GCF) is the largest factor that two or more terms share.
It is key in simplifying expressions because it allows us to reduce the complexity of the problem.
When working on a rational expression, finding the GCF helps in reducing both the numerator and the denominator.

For instance, in our exercise, we started by identifying the GCF of the numerator \(2x^2 - 5x\).
We found that both terms share \(x\) as their GCF.
In the denominator \(16x - 40\), the GCF was identified as \(8\).

By factoring out these GCFs, we obtain simpler forms of both the numerator and the denominator.
This makes it easier to identify common factors in the rational expression.
Eventually, this leads to a more straightforward simplification process.

Canceling common factors is the final step in simplifying rational expressions.
Once both the numerator and the denominator have been factored, we can look for any factors that appear in both.
These are the common factors.

In our example, after factoring, we had the expression: \(\frac{x(2x-5)}{8(2x-5)}\).
Notice that \((2x-5)\) is present in both the numerator and the denominator.
When we cancel out \((2x-5)\) from both, we are left with: \(\frac{x}{8}\).

Canceling common factors simplifies the rational expression to its lowest terms.
This step is crucial because it ensures the expression is as simple as possible and easier to work with in further calculations or applications.
Remember, you can only cancel factors, not terms.
It's important to factorize correctly before canceling anything.