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### Theory:

Excluded Value
A value that makes a rational expression (in its lowest form) undefined is called an Excluded value.
Suppose the rational expression $$\frac{p(x)}{q(x)}$$ is in its lowest form, then the value for which the expression becomes undefined is said to be its excluded value.

Working rule to find the excluded value of a rational number:
Step 1: Simplify or factorise the numerator $$p(x)$$ and the denominator $$p(x)$$.

Step 2: Cancel out the common factors in the numerator and the denominator.

Step 3: Equate the lowest form of the denominator $$q(x)$$ to zero.

Step 4: Thus, the obtained value for which the denominator becomes zero is the excluded value of that rational number.
Example:
Find the excluded value of the expression $$\frac{x^2 + 5x + 6}{(x + 2)(x - 5)}$$.

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.

$$\frac{x^2 + 5x + 6}{(x + 2))(x - 5)}$$ $$=$$ $$\frac{(x + 2)(x + 3)}{(x + 2))(x - 5)}$$

$$=$$ $\frac{\overline{)\left(x+2\right)}\left(x+3\right)}{\overline{)\left(x+2\right)}\left(x-5\right)}$

$$=$$ $$\frac{x + 3}{x - 5}$$

Step 3: Equate the lowest form of the denominator to zero.

$$x - 5$$ $$=$$ $$0$$

Add $$5$$ on both sides of the equation.

$$x - 5 + 5$$ $$=$$ $$0 + 5$$

$$\Rightarrow$$ $$x$$ $$=$$ $$5$$

Step 4: Write the excluded value.

The rational expression $$\frac{x^2 + 5x + 6}{(x + 2)(x - 5)}$$ is undefined when $$x$$ $$=$$ $$5$$.

That is $$\frac{x^2 + 5x + 6}{0}$$ $$=$$ not defined, when $$x$$ $$=$$ $$5$$.

Therefore, $$x$$ $$=$$ $$5$$ is called an excluded value for the rational expression $$\frac{x^2 + 5x + 6}{(x + 2)(x - 5)}$$.