Find the LCD for the given rational expression. [tex]$\frac{x+9}{4 x^2}, \frac{x-5}{7 x^2+14 x}$[/tex]
Answer (2)
Factor each denominator completely: 4 x 2 = 2 2 x 2 and 7 x 2 + 14 x = 7 x ( x + 2 ) .
Identify the unique factors: 2 , x , 7 , ( x + 2 ) .
Determine the highest power of each unique factor: 2 2 , x 2 , 7 1 , ( x + 2 ) 1 .
Multiply the highest powers to find the LCD: 28 x 2 ( x + 2 ) .
Explanation
Understanding the Problem We are given two rational expressions: 4 x 2 x + 9 and 7 x 2 + 14 x x − 5 . Our goal is to find the least common denominator (LCD) of these two expressions.
Factoring the Denominators First, we need to factor each denominator completely. The first denominator is 4 x 2 , which can be written as 2 2 x 2 . The second denominator is 7 x 2 + 14 x . We can factor out 7 x from this expression to get 7 x ( x + 2 ) .
Identifying Unique Factors Now, we identify the unique factors in both denominators: 2 , x , 7 , and ( x + 2 ) . We need to determine the highest power of each unique factor present in either denominator.
Determining Highest Powers The highest power of 2 is 2 2 = 4 . The highest power of x is x 2 . The highest power of 7 is 7 1 = 7 . The highest power of ( x + 2 ) is ( x + 2 ) 1 = ( x + 2 ) .
Calculating the LCD To find the LCD, we multiply these highest powers together: 4 ⋅ 7 ⋅ x 2 ⋅ ( x + 2 ) = 28 x 2 ( x + 2 ) . Therefore, the least common denominator is 28 x 2 ( x + 2 ) .
Final Answer Thus, the least common denominator (LCD) for the given rational expressions is 28 x 2 ( x + 2 ) .
Examples
When adding or subtracting fractions with polynomial expressions in the denominator, finding the least common denominator (LCD) is crucial. For example, if you are combining two circuits with different impedance formulas represented as rational expressions, determining the LCD helps simplify the overall impedance calculation. This is also applicable in mixture problems in chemistry, where you might need to combine different solutions with concentrations expressed as rational functions.
To find the least common denominator for the given rational expressions, we first factor each denominator. The unique factors and their highest powers lead us to the least common denominator, which is 28 x 2 ( x + 2 ) .
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