Adding and Subtracting Rational Expressions

We know that while adding or subtracting fractions, it is easy to do the operations when we have fractions with same denominators. Such fractions are called like fractions. For adding such fractions we simply add the numerators and put the new numerator over the common denominator.

This same method applies to rational-expressions as well. Addition and subtraction of rational expressions is not much different from addition and subtraction of fractional numbers. When adding and subtracting rational expressions (R. E.) with same denominators, all we do is add the numerators and put the new numerator over the common denominator.
Such rational-expressions are called like R. E. When adding the numerators also only the like terms are added the unlike terms are left as they are. Let us now try to understand this better using an example that requires us to add and subtract rational expressions.

Example 1: Simplify the following expression:
(2x+1)/x + (x-1)/x – (x^2-3)/x
Solution:
Note that here the denominators are same. Therefore these are like R. E. So all we need to do here is to add the numerators. Therefore,
(2x+1)/x + (x-1)/x – (x^2-3)/x
= [(2x+1) + (x-1) – (x^2-3)]/x
= [2x + 1 + x – 1 – x^2 + 3]/x
= [-x^2 + 3x + 2]/x Answer.
This is answer because this expression cannot be simplified any further.
Now we shall try to understand adding and subtracting rational expressions with unlike denominators using another example. When dealing with R. E. with unlike denominators we use the same approach that we use for fractions with unlike denominators.
First we find the LCD, next we convert each of the R. E. to an equivalent R. E. having same denominator and then add them like how we added R. E. with like denominators.

Example 2: Simplify
(2x+1)/x + (x-1)/x^2 – (x^2-3)/2
Solution:
(2x+1)/x + (x-1)/x^2 – (x^2-3)/2
Here LCD would be the lowest common multiple of the three terms x, x^2 and 2. Which is 2x^2. Therefore,
(2x+1)/x + (x-1)/x^2 – (x^2-3)/2
=[ (2x+1) * 2x]/[x*2x] + [(x-1)*2]/[x^2 * 2] – [(x^2-3)*x^2]/[2*x^2]
= [(4x^2+2x)/2x^2] + [(2x-2)/2x^2] – [(x^4-3x^2)/2x^2]
= [(4x^2+2x) + (2x-2) - (x^4-3x^2)]/2x^2
= [4x^2+2x + 2x-2 – x^4 + 3x^2]2x^2
= [-x^4 +7x^2 + 4x + 2]/2x^2 <- Answer.



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