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Chapter 7 Rational Expressions and Equations
Section 4 Finding the Least Common Denominator and Forming Equivalent Rational Expressions

Section 7.4 Objectives 1 Find the Least Common Denominator of Two or More Rational Expressions 2 Write a Rational Expression That Is Equivalent to a Given Rational Expression 3 Use the LCD to Write Equivalent Rational Expressions

Finding the LCD If the denominators of a sum or difference of rational expressions are not the same, the rational expressions must be written using a least common denominator. The least common denominator (LCD) of two or more rational expressions is the polynomial of least degree that is a multiple of each denominator in the expressions. After factoring the denominators, we can see that the LCD is (2)(3)(3 – w) = 6(3 – w).

Finding the LCD Finding the Least Common Denominator
Step 1: Factor each denominator completely. When factoring, write the factored form using powers. For example, write x2 + 4x + 4 as (x + 2)2. Step 2: If the factors are common except for their power, then list the factor with the highest power. That is, list each factor the greatest number of times that it appears in any one denominator. Then list the factors that are not common. Step 3: The LCD is the product of the factors written in Step 2.

Finding the LCD Example: Find the LCD of the rational expressions .
Factor each denominator. Use the factor that is repeated the greatest number of times.

Finding Equivalent Expressions
Steps to Form Equivalent Rational Expressions Step 1: Write each denominator in factored form. Step 2: Determine the “missing factor(s).” That is, what factor(s) does the new denominator have that is missing from the original denominator? Step 3: Multiply the original rational expression by Step 4: Find the product. Leave the denominator in factored form.

Finding Equivalent Expressions
Example: Write as an equivalent fraction with a denominator of 48. We want to change the denominator of 8 into a denominator of 48. We know that 8 · 6 = 48, so we form the factor of 1 = The equivalent fraction is

Finding Equivalent Expressions
Example: Write the rational expression with a denominator of x2y2z. We want to change the denominator of xyz to a denominator of x2y2z. We know that xyz · xy = x2y2z, so we form the factor of 1 = The equivalent fraction is

Writing Equivalent Expressions
Example: Find the LCD of the rational expressions Rewrite each expression. x2 + x = x(x + 1) The LCD = x(x + 1)(x – 2). x2 – x – 2 = (x – 2)(x + 1)

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