Rational Expressions and Restrictions

What you’ll learn

Simplify algebraic fractions without losing excluded values.

សម្រួលប្រភាគពិជគណិតដោយរក្សាតម្លៃដែលត្រូវដកចេញ។

Meaning and use

A rational expression is a ratio of polynomials. Factor before cancelling, and cancel only common multiplicative factors. Values that made an original denominator zero remain excluded even after simplification. An equivalent simplified form must carry those restrictions.

ប្រភាគពិជគណិតត្រូវបំបែកជាតួគុណមុនកាត់។ អាចកាត់តែកត្តារួម។ តម្លៃធ្វើឱ្យភាគបែងដើមសូន្យនៅតែត្រូវដកចេញ ទោះសម្រួលហើយក៏ដោយ។

Formulas

Common factors

\[\frac{ab}{ac}=\frac bc\quad(a\ne0,\ c\ne0)\]

Factoring a difference of squares

\[\frac{x^2-9}{x-3}=x+3\quad(x\ne3)\]

Symbols and units

Symbol	Meaning	Unit or restriction
a, b, c	Factors in a rational expression	Denominator factors must be nonzero
x	Variable	Subject to original denominator restrictions

Conditions and limits

• Record all original denominator exclusions before cancelling.
• Never cancel terms across addition or subtraction unless they have first become a common factor.
• ការសម្រួលមិនអាចស្ដារតម្លៃដែលភាគបែងដើមសូន្យវិញទេ។

Worked examples

Cancelling a common factor

Simplify \(\frac{6x}{3x}\).

1. Original denominator requires \(x\ne0\).
2. Cancel 3x to obtain 2.

2, with x ≠ 0

ទោះចម្លើយសាមញ្ញជា 2 ក៏ x=0 នៅតែមិនអនុញ្ញាត។

Factoring before cancellation

Simplify \(\frac{x^2-9}{x-3}\).

1. Factor numerator \((x-3)(x+3)\).
2. Cancel x − 3 only when \(x\ne3\).

\(x+3\), with x ≠ 3

បំបែកជាផលគុណមុនកាត់ ហើយរក្សាលក្ខខណ្ឌដើម។

Common mistakes

Instead of	Use	Why
(x + 2)/x = 2.	(x + 2)/x = 1 + 2/x, with x ≠ 0.	x is not a factor of the whole numerator.
(x² − 9)/(x − 3) = x + 3 for all x.	The simplification requires x ≠ 3.	The original denominator is zero at 3.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. Simplify 8x/(4x) and state the restriction.
2. Which x is excluded from 1/(x + 5)?
3. Simplify (x² − 4)/(x − 2).

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 2, with x ≠ 0

   Cancel the common factor only on the original domain.

2. x = −5

   It makes the denominator zero.

3. x + 2, with x ≠ 2

   Factor x² − 4 as (x − 2)(x + 2).

Quick recap

• Factor before cancelling.
• Cancel factors, not added terms.
• Keep original domain restrictions.