describe the mathematic process of canceling like factors when working with rational expressions and demonstrate this with an example
When working with rational expressions, canceling like factors involves simplifying the expression by dividing both the numerator and the denominator by their common factors. This process is similar to canceling common factors when simplifying fractions.
To cancel like factors in a rational expression, you need to factor both the numerator and the denominator, identifying the common factors. Then, you can divide both the numerator and the denominator by those common factors to simplify the expression.
Let's demonstrate this process with an example:
Consider the rational expression: (6x^2 - 12x) / (3x)
First, factor the numerator and denominator:
Numerator: 6x^2 - 12x = 6x(x - 2)
Denominator: 3x = 3 * x
Now, identify the common factors in the numerator and denominator. In this case, the common factor is the variable "x".
Next, cancel out the common factor by dividing both the numerator and the denominator by "x":
(6x(x - 2)) / (3 * x) = (6(x - 2)) / (3)
Finally, simplify the expression further if possible:
(6(x - 2)) / 3 = 2(x - 2)
So, by canceling the like factor "x" in the original rational expression, we simplified it to 2(x - 2).