Find the LCD 5/(2(x+1)) , 9/(2x) , 1/4

Solution:

Step 1:

To determine the Least Common Denominator (LCD) for the fractions $\frac{5}{2(x+1)}$, $\frac{9}{2x}$, and $\frac{1}{4}$, we must first find the Least Common Multiple (LCM) of their denominators: $2(x+1)$, $2x$, and $4$.

Step 2:

To find the LCM of $2(x+1)$, $2x$, and $4$, we will break it down into three parts: the coefficients (numerical part), the variables (literal part), and the expressions involving variables (compound part). We will then calculate the LCM for each part and multiply them to get the final LCM.

Step 3:

The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers. To find it:

1. Decompose each number into its prime factors.

2. For each prime factor, take the highest power that appears in the factorization of any of the numbers.

Step 4:

The number $2$ is already a prime number with no other factors besides $1$ and $2$.

Step 5:

The number $4$ can be factored into $2\times 2$.

Step 6:

The LCM of the numerical part is $4$, obtained by multiplying $2$ by $2$.

Step 7:

For the variable $x$, which is raised to the first power, the factor is $x$ itself, appearing once.

Step 8:

The LCM of the variable part $x$ is simply $x$, as it is the highest power of $x$ present.

Step 9:

For the compound variable part $x+1$, the factor is the expression itself, $x+1$, appearing once.

Step 10:

The LCM of the compound variable part $x+1$ is the expression $x+1$, as it does not share common factors with the other terms.

Step 11:

The final LCM, or LCD, of the denominators is the product of the LCMs of the numerical, variable, and compound parts: $4x(x+1)$.

Knowledge Notes:

The Least Common Denominator (LCD) is the least common multiple of the denominators of a set of fractions. It is used to find a common denominator for the fractions to simplify addition, subtraction, or comparison.

The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers. It is important in finding the LCD because the LCD is the LCM of the denominators.

Prime Factorization is the process of decomposing a number into a product of prime numbers. This is useful in finding the LCM because the LCM is calculated by multiplying the highest powers of all prime factors involved.

When dealing with algebraic expressions as part of the LCM, each unique algebraic term is treated as a separate entity, and its highest power is used in the LCM.

The LCM of algebraic expressions involving variables is found by taking the highest power of each variable that appears in any of the expressions. If an expression is a compound expression (like $x+1$), it is treated as a unique factor unless it can be factored further into a product of simpler expressions.