WHY rational expressions that are being added must have an LCD, but rational expressions that are being multiplied do not need to have an LCD.
The short answer is that addition and multiplication are different operations with different properties.
If we added rational expressions directly without using the LCD the answer would be nonsense. Consider this
1/2 + 1/2 = (1+1)/(2+2) = 2/4 = 1/2 ??
Does that make any sense?
We know the right answer should be 1/2 + 1/2 = 1.
To understand why rational expressions that are being added must have a least common denominator (LCD), but rational expressions that are being multiplied do not need to have an LCD, let's start by explaining what a rational expression is.
A rational expression is an expression that represents the ratio of two polynomials, where the denominator is not equal to zero. For example, 3/(x+1) and (4x+2)/(x^2+5x+6) are rational expressions.
When adding rational expressions, we need to ensure that they have a common denominator so that we can combine them correctly. The common denominator is the least common multiple (LCM) of the denominators of the individual rational expressions. This is necessary because we cannot directly add fractions with different denominators. By finding the LCD, we can rewrite each rational expression with an equivalent fraction that has the common denominator.
For example, let's consider adding 1/(x+1) and 1/(x+2). The denominators are x+1 and x+2, respectively. To find the LCD, we need to determine the least common multiple of x+1 and x+2. Since there are no common factors between them, the LCD is simply the product of the two denominators, which is (x+1)(x+2).
Now, we can rewrite each rational expression with the LCD:
1/(x+1) = (x+2)/(x+1)(x+2)
1/(x+2) = (x+1)/(x+1)(x+2)
With the common denominator, we can safely add the rational expressions:
1/(x+1) + 1/(x+2) = (x+2)/(x+1)(x+2) + (x+1)/(x+1)(x+2)
= (x+2 + x+1)/(x+1)(x+2)
= (2x + 3)/(x+1)(x+2)
On the other hand, when multiplying rational expressions, we can simply multiply the numerators together and the denominators together without the need for a common denominator. The resulting expression will be the product of the two original expressions.
For example, let's consider multiplying 3/(x+1) and (x+2)/5. To find the product, we multiply the numerators and the denominators:
(3/(x+1)) * ((x+2)/5) = (3(x+2))/((x+1)*5)
= (3(x+2))/(5(x+1))
As you can see, we don't need to find a common denominator in this case. We simply multiply the numerators and the denominators separately.
In summary, when adding rational expressions, we need to find the least common denominator to ensure the fractions have a common base for addition. However, when multiplying rational expressions, we can directly multiply the numerators and denominators without the need for a common denominator. The distinction is due to the different properties of addition and multiplication with fractions.