### 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**:

*: Factorise the numerator \(x^2 + 5x + 6\) by splitting the middle term.*

**Step 1**\(x^2 + 5x + 6\) \(=\) \(x^2 + 2x + 3x + 6\)

\(=\) \(x (x + 2) + 3 (x + 2)\)

\(=\) \((x + 2)(x + 3)\)

*: Rewrite the expression and cancel out the common factors.*

**Step 2**\(\frac{x^2 + 5x + 6}{x + 2}\) \(=\) \(\frac{(x + 2)(x + 3)}{x + 2}\)

\(=\) $\frac{\overline{)\left(x+2\right)}\left(x+3\right)}{\overline{)\left(x+2\right)}}$

\(=\) \(x + 3\)

*: Write the rational expression in its lowest form.*

**Step 3**\(\frac{x^2 + 5x + 6}{x + 2}\) \(=\) \(x + 3\)

Therefore, the rational expression \(\frac{x^2 + 5x + 6}{x + 2}\) is reduced to \(x + 3\).