Simplifying Rational Expressions

Selasa, 25 Juli 2023

A fraction denotes a quotient. In algebra, the fraction  is called a rational expression. In a rational expression, both the numerator and denominator are polynomials.

Every polynomial is a rational expression because it can be written as a quotient with 1 in the denominator.

 

Zero cannot be the denominator of a fraction because division by zero is undefined. In the expression , if x = –3, the denominator equals zero. So, any value assigned to a variable that results in a denominator of zero must be excluded from the domain of the variable. These values are called excluded values.

 

A function that contains a rational expression is called a rational function. You can use the graph of the rational function of a rational expression to investigate excluded values of the variable.

Example

Find the excluded value(s) for each rational expression.

 

 

Alternative Solutions:

 

Exclude the values for which 2 + m = 0.

2 + m = 0

 m = –2

So, m cannot equal –2.

 

 

Alternative Solutions:

 

Exclude the values for which a(a – 4) = 0.

a(a – 4) = 0

a = 0 or   a – 4 = 0          Zero Product Property

So, a cannot equal 0 or 4.

 

 

Alternative Solutions:

 

Exclude the values for which n² – 25 = 0.

n² – 25 = 0

(n + 5)(n – 5) = 0                      Factor n² – 25

n = –5          or       n = 5           Zero Product Property

So, a cannot equal –5 or 5.

 

Recall that you can simplify a fraction by using the following steps.

·        First, factor the numerator and denominator.

·        Then, divide the numerator and denominator by the greatest common factor.

 

Example

History Link

 

4.   In the 1960 presidential election, more than 60% of the registered voters cast ballots. No presidential election since 1960 has had a greater voter turnout. Express 60% as a fraction in simplest form.

 

Alternative Solutions:

You can use the same procedure to simplify rational expressions that have polynomials in the numerator and denominator. To simplify means that the numerator and denominator have no factors in common, except 1.

 

Example

Simplify each rational expression.

 

 

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Labels: Mathematician

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