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Theory:

Rational Expression
An expression is called a rational expression if it can be written in the form \(\frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomials and \(q(x) \neq 0\).
 
A rational expression is the ratio of two polynomials.
Example:
The following are few examples of rational expressions.
 
1. \(\frac{x + y}{x - y}\), where \(x \neq y\)
 
2. \(\frac{3ab}{5a^2b^3}\)
 
3. \(\frac{x^2 + 3x}{x^5}\)
Reduction of Rational Expression
A rational expression \(\frac{p(x)}{q(x)}\) is said to be in its lowest form if the greatest common divisor of \(p(x)\) and \(q(x)\) is \(1\).
 
That is \(GCD \left(p(x), q(x) \right)\) = \(1\).
Working rule to reduce a rational expression to its lowest form:
Step 1: Simplify or factorise the numerator \(p(x)\) and the denominator \(q(x)\).
 
Step 2: Cancel out the common factors in the numerator and the denominator.
 
Step 3: The final expression obtained after the above two steps is the rational expression in its lowest form.
Example:
Reduce the expression \(\frac{x^2 + 5x + 6}{x + 2}\).
 
Solution:
 
Step 1: Factorise the numerator \(x^2 + 5x + 6\) by splitting the middle term.
 
\(x^2 + 5x + 6\) \(=\) \(x^2 + 2x + 3x + 6\)
 
\(=\) \(x (x + 2) + 3 (x + 2)\)
 
\(=\) \((x + 2)(x + 3)\)
 
Step 2: Rewrite the expression and cancel out the common factors.
 
\(\frac{x^2 + 5x + 6}{x + 2}\) \(=\) \(\frac{(x + 2)(x + 3)}{x + 2}\)
 
\(=\) x+2x+3x+2
 
\(=\) \(x + 3\)
 
Step 3: Write the rational expression in its lowest form.
 
\(\frac{x^2 + 5x + 6}{x + 2}\) \(=\) \(x + 3\)
 
Therefore, the rational expression \(\frac{x^2 + 5x + 6}{x + 2}\) is reduced to \(x + 3\).