Find the LCD (4x)/(x-3) and x/(x-5)
The question is asking for the calculation of the Least Common Denominator (LCD) for the two given rational expressions. The first expression is (4x)/(x-3) and the second expression is x/(x-5). The LCD is the smallest expression that both denominators can divide into without leaving a remainder, and it is used to combine these fractions through addition or subtraction by providing a common basis for the denominators. To find the LCD, the question hints at identifying and possibly factoring the denominators (if they are factorable), then combining these factors to determine the smallest expression that is divisible by each original denominator.
$\frac{4 x}{x - 3}$and$\frac{x}{x - 5}$
Answer
Solution:
Step 1:
Identify the Least Common Denominator (LCD) by determining the Least Common Multiple (LCM) of the denominators \( x - 3 \) and \( x - 5 \).
Step 2:
To find the LCM, follow these steps: a. Decompose each term into its prime factors.
b. For each prime factor, take the highest power that appears in any of the terms.
Step 3:
Note that the number \( 1 \) is not considered a prime number because it has only one divisor, itself. Thus, it is not included in prime factorization.
Step 4:
Since \( 1 \) is not a prime, the LCM of \( 1 \) and \( 1 \) is simply the product of the highest powers of prime factors found in the terms, which is \( 1 \).
Step 5:
The expression \( x - 3 \) is a non-factorable polynomial and is treated as a unique factor. It appears once as \( (x - 3) = x - 3 \).
Step 6:
Similarly, \( x - 5 \) is also a non-factorable polynomial and is considered as a unique factor. It appears once as \( (x - 5) = x - 5 \).
Step 7:
The LCM of \( x - 3 \) and \( x - 5 \) is the product of these unique factors taken to the highest power they occur in either expression, which results in \( (x - 3)(x - 5) \).
Knowledge Notes:
Least Common Denominator (LCD): The LCD of two or more fractions is the smallest number that can be used as a common denominator for the fractions. It is equivalent to the Least Common Multiple (LCM) of the denominators.
Least Common Multiple (LCM): The LCM of two or more expressions is the smallest expression that is a multiple of all the expressions. For polynomials, the LCM is found by multiplying each distinct factor the greatest number of times it occurs in any of the expressions.
Prime Factorization: This is the process of breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number. Prime factorization is used to find the LCM of numerical values.
Non-factorable Polynomials: These are polynomials that cannot be factored into the product of two or more non-constant polynomials with integer coefficients. In the context of finding the LCM, such polynomials are treated as unique factors.
Unique Factors: In the context of LCM, unique factors refer to factors that are not shared between the expressions. Each unique factor is included in the LCM once, raised to the highest power with which it appears in any of the expressions.
