In this lesson you’ll learn what rational expressions are and how to simplify them by factoring the numerator and denominator completely.
A rational expression is a fraction where the numerator and/or denominator is a polynomial.
To simplify a rational expression:
Factor the numerator completely.
Factor the denominator completely.
Cancel any common factors (things that are multiplied in both).
Write the simplified expression in lowest terms.
Factor, then cancel common factors Original: x² − 9 x² − 6x + 9 → Factor: (x − 3) (x + 3) (x − 3) (x − 3) = x + 3 x − 3 The red (x − 3) factors cancel because they appear in both numerator and denominator. Excluded value: x ≠ 3 (would make the original denominator zero). Remember: you can only cancel factors (things multiplied), never terms (things added).
Important rules:
You can only cancel factors (things multiplied), never terms that are added or subtracted.
Always note the values that make any denominator zero. These are excluded from the domain.
1. Simplify (x² − 9) / (x² − 6x + 9)
Factor both:
x 2 − 9 x 2 − 6 x + 9 = ( x − 3 ) ( x + 3 ) ( x − 3 ) ( x − 3 ) \frac{x^2 - 9}{x^2 - 6x + 9} = \frac{(x - 3)(x + 3)}{(x - 3)(x - 3)} x 2 − 6 x + 9 x 2 − 9 = ( x − 3 ) ( x − 3 ) ( x − 3 ) ( x + 3 ) Cancel the common (x − 3):
= x + 3 x − 3 ( x ≠ 3 ) = \frac{x + 3}{x - 3} \qquad (x \neq 3) = x − 3 x + 3 ( x = 3 ) 2. Simplify (6x² − 12x) / (3x² − 6x)
Factor both by pulling out the GCF:
6 x ( x − 2 ) 3 x ( x − 2 ) = 6 3 = 2 ( x ≠ 0 , x ≠ 2 ) \frac{6x(x - 2)}{3x(x - 2)} = \frac{6}{3} = 2 \qquad (x \neq 0,\; x \neq 2) 3 x ( x − 2 ) 6 x ( x − 2 ) = 3 6 = 2 ( x = 0 , x = 2 ) 3. Simplify (x² + 5x + 6) / (x² − 4)
Factor both:
( x + 2 ) ( x + 3 ) ( x − 2 ) ( x + 2 ) = x + 3 x − 2 ( x ≠ − 2 , x ≠ 2 ) \frac{(x + 2)(x + 3)}{(x - 2)(x + 2)} = \frac{x + 3}{x - 2} \qquad (x \neq -2,\; x \neq 2) ( x − 2 ) ( x + 2 ) ( x + 2 ) ( x + 3 ) = x − 2 x + 3 ( x = − 2 , x = 2 ) Simplifying rational expressions is useful when:
Finding average cost per unit (total cost / number of units)
Simplifying rates in work problems
Reducing formulas in physics or engineering
Working with concentrations in mixtures
Example: if total cost is 2x² + 10x + 50 and you produce x items, average cost per item is (2x² + 10x + 50) / x. Simplifying helps analyze efficiency as production increases.
Simplifying rational expressions is mostly about strong factoring skills. Factor everything completely first, then cancel matching factors top and bottom. Always write down the excluded values (where the denominator would be zero). With practice, this process becomes fast and reliable.
Simplify $\frac{x^2 - 4}{x - 2}$ A. $x - 2$ B. $x + 2$ C. $x + 4$ D. $x^2 - 2$
When simplifying rational expressions, you can cancel: A. Only the leading coefficients B. Common terms that are added C. Only constants D. Common factors
Simplify $\frac{6x^2 - 12x}{3x^2 - 6x}$ A. $2$ B. $2x$ C. $x - 2$ D. $2(x - 2)$
Why do we note excluded values when simplifying? A. Because they are always roots B. To make the expression look nicer C. They make the original denominator zero D. To avoid fractions
Simplify $\frac{x^2 + 5x + 6}{x^2 + 2x}$. A. $\frac{x + 2}{x}$ B. $\frac{x + 3}{x}$ C. $\frac{x + 6}{2x}$ D. $\frac{5x + 6}{2x}$
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