Find the LCD for the following rational expressions.
[tex]\frac{2 x-7}{10 x^2-2 x}, \frac{7 x+1}{27 x^2+9 x}[/tex]
[tex]LCD = \square[/tex] (Simplify your answer.)
Answer (2)
Factor the denominators: 10 x 2 − 2 x = 2 x ( 5 x − 1 ) and 27 x 2 + 9 x = 9 x ( 3 x + 1 ) .
Identify the unique factors: 2 , x , ( 5 x − 1 ) , 9 , and ( 3 x + 1 ) .
Multiply the unique factors with the highest powers: 2 ⋅ 9 ⋅ x ⋅ ( 5 x − 1 ) ⋅ ( 3 x + 1 ) .
Simplify to find the LCD: 18 x ( 5 x − 1 ) ( 3 x + 1 ) .
Explanation
Understanding the Problem We are given two rational expressions: 10 x 2 − 2 x 2 x − 7 and 27 x 2 + 9 x 7 x + 1 . Our goal is to find the least common denominator (LCD) of these two rational expressions.
Plan of Action To find the LCD, we first need to factor the denominators of both rational expressions completely.
Factoring the First Denominator Let's factor the first denominator, 10 x 2 − 2 x . We can factor out a 2 x from both terms: 10 x 2 − 2 x = 2 x ( 5 x − 1 ) .
Factoring the Second Denominator Now, let's factor the second denominator, 27 x 2 + 9 x . We can factor out a 9 x from both terms: 27 x 2 + 9 x = 9 x ( 3 x + 1 ) .
Identifying Unique Factors Now we identify the unique factors from both denominators. The factors are 2 , x , ( 5 x − 1 ) , 9 , and ( 3 x + 1 ) .
Determining the LCD The LCD is the product of the highest powers of all unique factors present in the denominators. So, we have 2 , 9 , x , ( 5 x − 1 ) , and ( 3 x + 1 ) . Since 2 and 9 are both constants, we multiply them together: 2 × 9 = 18 . Therefore, the LCD is 18 x ( 5 x − 1 ) ( 3 x + 1 ) .
Final Answer Thus, the LCD for the given rational expressions is 18 x ( 5 x − 1 ) ( 3 x + 1 ) .
Examples
When adding or subtracting fractions with polynomial denominators, finding the LCD is essential. For example, if you are combining data from two different sources where the data is represented as rational functions, you need to find the LCD to combine the data correctly. This is similar to adding fractions like 6 1 and 8 1 , where you need to find the LCD, which is 24, to add them: 6 1 + 8 1 = 24 4 + 24 3 = 24 7 . The same principle applies to rational expressions.
To find the LCD of the expressions 10 x 2 − 2 x 2 x − 7 and 27 x 2 + 9 x 7 x + 1 , we factor their denominators to get 2 x ( 5 x − 1 ) and 9 x ( 3 x + 1 ) . The unique factors lead us to the LCD, which simplifies to 18 x ( 5 x − 1 ) ( 3 x + 1 ) .
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