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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{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)}.
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