Simplify (x+2)/(3x-6)

Solution:

Step 1:

Extract the factor of $3$ from the term $3x$. $\frac{x+2}{3(x-2)}$

Step 2:

Similarly, take out the factor of $3$ from the term $-6$. $\frac{x+2}{3(x-2)}$

Step 3:

Rewrite the denominator by combining the factored out $3$. $\frac{x+2}{3(x-2)}$

Knowledge Notes:

To simplify the expression $\frac{x+2}{3x-6}$, we need to factor common terms in the numerator and the denominator. The goal is to identify and cancel out any common factors to simplify the expression to its lowest terms.

Relevant knowledge points include:

1. Factoring: This is the process of breaking down an expression into a product of simpler expressions. In this case, we identify that $3$ is a common factor in the terms of the denominator.

2. Simplifying Rational Expressions: A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying a rational expression often involves factoring both the numerator and the denominator and then canceling out common factors.

3. Distributive Property: This property is used when factoring out a common factor from an expression. It states that $a(b+c)=ab+ac$. We use this property in reverse to factor out the $3$ from $3x-6$ to get $3(x-2)$.

In the given problem, the denominator $3x-6$ can be factored by taking out the common factor of $3$, resulting in $3(x-2)$. Since there are no common factors in the numerator $x+2$ and the factored denominator $3(x-2)$, the expression is already simplified, and no further cancellation is possible.

The final simplified form of the expression is $\frac{x+2}{3(x-2)}$.