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Addition and Subtraction of Rational Expressions

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Fractional expressions that have a polynomial in either the numerator or the denominator are called rational expressions. The word “rational” refers to “fraction”. The fractions are numbers are depicted in the form of “p/q” and q ≠ 0, where the upper part of the fraction, “p,” is the numerator, and the lower part of the fraction, “q,” is the denominator. Hence, all rational expressions must be fractions. Rational expressions are also known as rational polynomials. Some examples of rational expressions are 2/x, (x-y/3x), (a+3b/2a-b), etc. We can perform various operations on rational expressions just like fractions. So, to add and subtract rational expressions, we use the same rules for adding and subtracting fractions as for adding and subtracting just numbers. In this article, we will learn how to add and subtract rational expressions.

How to Add and Subtract Rational Expressions?

For adding and subtracting rational expressions, go through the following steps:

  1. Factorize the denominator.
  2. Find the least common multiple of the denominator and rewrite each fraction with the common denominator.
  3. Now, add or subtract the numerators of the two rational expressions.
  4. If possible, then factorize again.
  5. Write the final answer in its simplified form.

Let’s discuss an example, to understand the concept better.

Example: Solve: (6a+5b)/9a + (4a−3b)/9a.

Solution:

Given expression: (6a+5b)/9a + (4a−3b)/9a.

Step 1: Let us check whether there is a possibility of factorization of the denominator. When both the terms in an expression can be divided by the same number or a variable, we can perform factorization. Here, the problem does not require any factorization.

Step 2: Before proceeding with the addition of both fractions, they must have a common denominator, which means the denominators of both fractions must be the same. Here, both denominators are the same, which implies that both fractions have a common denominator. So, no changes need to be made to the problem in this step.

Step 3: As we have a common denominator, we can proceed to the next step, i.e., combining the numerators and placing the combination over the common denominator. Make sure that the second expression is always placed in a set of parenthesis.

(6a+5b)/9a + (4a−3b)/9a = [(6a+5b)+(4a−3b)]/9a

Since the operation for this problem is addition, add the like terms of both the numerators. The like terms are (6a, 4a) and (5b, -3b).

[(6a+5b)+(4a−3b)]/9a = (10a+2b)/9a = 2(5a+b)/9a

Step 4: Now, the next step is to determine whether any factorization is possible. As we learned earlier, we can perform factorization when both the terms in an expression can be divided by the same number or a variable. Here, the obtained result does not require any additional factoring. Hence, the final answer is,

(6a+5b)/9a + (4a−3b)/9a = 2(5a+b)/9a

Problems based on Addition and subtraction of Rational Expression

Problem 1: Solve 5/(3y+4) +(y−6)/(3y + 4). 

Solution: 

Given expression: 5/(3y+4) +(y−6)/(3y + 4)

We can observe that both fractions have a common denominator.

5/(3y+4) +(y−6)/(3y + 4) = {5 + (y – 6)}/(3y + 4)

= (y – 1)/(3y + 4)

Thus, 5/(3y+4) +(y−6)/(3y + 4) = (y – 1)/(3y + 4).

Problem 2: Solve (2x+7)/(x+5) − (6x−1)/(x−2).

Solution: 

Given expression: (2x+7)/(x+5) − (6x−1)/(x−2).

We can observe that both fractions don’t have a common denominator.

So, find the least common multiple of the denominators, i.e., (x+5) and (x−2): (x+5)(x−2)

Now, (2x+7)/(x+5) − (6x − 1)/(x-2)

= [(2x+7)(x−2)/(x+5)(x−2)] − [(6x−1)(x+5)/(x+5)(x−2)]

= [(2x2−4x+7x−14)/(x+5)(x−2)] − [(6x2+30x−x−5)/(x+5)(x−2)]

= [(2x2+3x-14)/(x+5)(x−2)] − [(6x2+29x−5)/(x+5)(x−2)] 

= [(2x2+3x-14) − (6x2+29x−5)]/(x+5)(x−2)

= (−4x2−26x−9)/(x+5)(x−2)

= −(4x2+26x+9)/(x+5)(x−2)

Hence, (2x+7)/(x+5) − (6x−1)/(x−2) = −(4x2+26x+9)/(x+5)(x−2).

Problem 3: Solve 4x/(7x−2) + (19−2x)/(7x−2).

Solution:

Given expression: 4x/(7x−2) + (19−2x)/(7x−2).

We can observe that both fractions have a common denominator.

4x/(7x−2) + (19 – 2x)/(7x−2) = (4x + 19−2x)/(7x−2)

= (2x+19)/(7x−2)

Hence, 4x/(7x−2) + (19−2x)/(7x−2) = (2x+19)/(7x−2).

Problem 4: Solve 8/(x−3) − 4x/(x2−9).

Solution:

Given expression: 8/(x−3) − 4x/(x2−9).

We can observe that both fractions don’t have a common denominator.

(x2−9) = (x−3)(x+3)   {Since, a2 − b2 = (a−b)(a + b)}

So, find the least common multiple of the denominators, i.e., (x−3) and (x−3)(x+3): (x−3)(x+3)

8/(x−3)−4x/(x2−9) = [8(x+3)/(x−3)(x+3)] − [4x/(x−3)(x+3)]

= [8x+24/(x2−9)] − [4x/(x2−9)]

= (8x+24−4x)/(x2−9)

= (4x+24)/(x2−9)

= 4(x+6)/(x2−9)

Hence, (8/x−3)−(4x/x2−9) = 4(x+6)/(x2−9).

Problem 5: Solve (6u−v/7u) + (3u+5v/8v).

Solution:

Given expression: (6u−v/7u) + (3u+5v/8v).

We can observe that both fractions don’t have a common denominator.

So, find the least common multiple of the denominators, i.e., 7u and 8v: 56uv.

(6u−v/7u) + (3u+5v/8v) = [(6u−v)(8v)/(7u)(8v)] − [(3u+5v)(7u)/(8v)(7u)]

= [(48uv−8v2)/56uv] − [(21u2+35uv)/56uv]

= [(48uv−8v2)−(21u2+35uv)/56uv]

= (−8v2+13uv−21u2)/56uv

= −(21u2−13uv+8v2)/56uv

Hence, (6u−v/7u) + (3u+5v/8v) = −(21u2−13uv+8v2)/56uv.



Last Updated : 26 Dec, 2023
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