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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.
 
\(\frac{x^2 + 5x + 6}{(x + 2))(x - 5)}\) \(=\) \(\frac{(x + 2)(x + 3)}{(x + 2))(x - 5)}\)
 
\(=\) x+2x+3x+2x5
 
\(=\) \(\frac{x + 3}{x - 5}\)
 
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 \(\frac{x^2 + 5x + 6}{(x + 2)(x - 5)}\).