Addition and Subtraction of Rational Expressions: Explanation

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$\frac{2 x^{2} - 1}{3 x + 4}$

In this article, we are going to see how to add and subtract Rational Expressions.

Addition and Subtraction of Rational Expressions

Recall how to add or subtract fractions together. Since Rational Expressions are essentially fractions whose numerator and denominator are both polynomials, we can use the same basic concept here.

The Golden Rule for Adding and Subtracting Fractions

The golden rule for adding and subtracting fractions together is:

If the fractions to be added or subtracted have the same denominators, the corresponding numerators can simply be added or subtracted keeping the denominator constant.

This essentially leads to two types of use cases:

Rational expressions with common denominators
Rational expressions with different denominators

Sum Rational Expressions with Common Denominators

To add or subtract expressions with like denominators, simply add or subtract (depending on the sign) the numerators keeping the common denominator constant.

Evaluate

$\frac{2 x^{3}}{5} + \frac{9 x^{3}}{5}$

Solution

Given that the denominators are common, we can add the numerators together, keeping the denominator constant.

$\frac{2 x^{3}}{5} + \frac{9 x^{3}}{5} = \frac{2 x^{3} + 9 x^{3}}{5} = \frac{11 x^{3}}{5}$

Evaluate

$\frac{x^{2}}{x - 1} - \frac{2 x - 1}{x - 1}$

Solution

$\frac{x^{2}}{x - 1} - \frac{2 x - 1}{x - 1} = \frac{x^{2} - (2 x - 1)}{x - 1} = \frac{x^{2} - 2 x + 1}{x - 1}$

Since x²-2x+1 is divisible by (x-1), we can further simplify the expression.

$\frac{x^{2} - 2 x + 1}{x - 1} = \frac{{(x - 1)}^{2}}{x - 1} = x - 1$

Be careful that the negative sign is distributed over the entire polynomial. So -(2x-1) = -2x+1. All terms in the polynomial need to be negated.

Sum Rational Expressions with Different Denominators

If we have to perform addition or subtraction on expressions with different denominators, we first manipulate the expressions so that they end up having the same denominators.

You can follow the following procedure for adding and subtracting fractions with unlike denominators:

Step 1: Replace the denominator of each term with the Lowest Common Multiple (LCM) of all the denominators.

Step 2: Replace the numerator of each term with $(o r i g i n a l n u m e r a t o r) \times \frac{L C M o f d e n o m i n a t o r s}{O r i g i n a l d e n o m i n a t o r}$ .

Step 3: Now that all the denominators are the same, add or subtract the numerators together to obtain the resultant numerator over the common denominator.

Step 4: Simplify the expression if necessary.

Since addition and subtraction of polynomials with unlike denominators involves calculating the LCM of the denominators, which are polynomials, it is necessary to be comfortable with calculating the LCM of given polynomials.

Finding the Lowest Common Multiple of Two Polynomials

The process of finding the LCM of algebraic polynomials is no different from finding the LCM of a given set of integers. Before moving on to examples of adding and subtracting polynomials with different denominators, let us first work through an example of evaluating the LCM of 2 polynomials.

Find the LCM of 15mn and 21np².

Solution:

Let us break up the terms into their prime factors and smallest variable factors.

15mn = 3×5×m×n

21np² = 3×7×n×p×p

Multiply each factor the greatest number of times it appears in any of the factorizations.

LCM = 3×5×7×m×n×p×p

=105mnp²

Now let us look at some examples of adding and subtracting rational expressions with unlike denominators.

Evaluate

$\frac{m + n}{2} + \frac{2 n + 1}{5}$

Solution

$\frac{m + n}{2} + \frac{2 n + 1}{5} = \frac{2 (m + n) + 5 (2 n + 1)}{10} (L C M o f t h e d e n o m i n a t o r s i s 10) = \frac{2 m + 2 n + 10 n + 5}{10} = \frac{2 m + 12 n + 5}{10}$

Evaluate

$\frac{25}{a} + \frac{11}{2 a} - \frac{35}{a^{2}}$

Solution

\frac{25}{a} + \frac{11}{2 a} - \frac{35}{a^{2}} = \frac{2 a \times 25 + a \times 11 - 35}{2 a^{2}} (L C M o f d e n o m i n a t o r s i s 2 a^{2}) = \frac{50 a + 11 a - 35}{2 a^{2}} = \frac{61 a - 35}{2 a^{2}}

Evaluate

$\frac{8}{4 x^{2} - y^{2}} - \frac{3}{2 x + y}$

Solution

In this example, we have the following polynomials in the denominator :

4x²-y² and 2x+y.

Now,

4x²-y²=(2x+y)(2x-y)

So the LCM of the denominators is (2x+y)(2x-y) [or 4x²-y²]

$\frac{8}{4 x^{2} - y^{2}} - \frac{3}{2 x + y} = \frac{8 - (2 x - y) 3}{(2 x + y) (2 x - y)} = \frac{8 - 6 x + 3 y}{(2 x + y) (2 x - y)}$

Addition and Subtraction of Rational Expressions - Key takeaways

A rational expression is an algebraic fraction whose numerator and denominator are both polynomials.
To add or subtract expressions with like denominators, simply add or subtract (depending on the sign) the numerators, keeping the common denominator constant.
If we have to perform addition or subtraction on expressions with different denominators, we first manipulate the expressions so that they end up having the same denominators.

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Frequently Asked Questions about Addition and Subtraction of Rational Expressions

How do you solve the addition and subtraction of rational algebraic expressions?

How to solve the addition and subtraction of rational algebraic expressions: If the expressions to be added or subtracted have the same denominators, the corresponding numerators can simply be added or subtracted keeping the denominator constant. If they have different denominators, we take the LCM of the denominators and multiply each numerator with the appropriate factor and proceed to add or subtract them.

What is addition and subtraction of rational expression?

A rational expression is an algebraic fraction whose numerator and denominator are both polynomials. We can add or subtract multiple rational expressions together.

What are the rules for adding and subtracting rational expression?

If the expressions to be added or subtracted have the same denominators, the corresponding numerators can simply be added or subtracted keeping the denominator constant. If they have different denominators, we take the LCM of the denominators and multiply each numerator with the appropriate factor and proceed to add or subtract them.

How do you add or subtract rational expressions with different denominators?

If rational expressions to be added/subtracted have different denominators, we take the LCM of the denominators and multiply each numerator with the appropriate factor and proceed to add or subtract these updated numerators.

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Addition and Subtraction of Rational Expressions

StudySmarter Editorial Team

Addition and Subtraction of Rational Expressions

The Golden Rule for Adding and Subtracting Fractions

Sum Rational Expressions with Common Denominators

Sum Rational Expressions with Different Denominators

Finding the Lowest Common Multiple of Two Polynomials

Addition and Subtraction of Rational Expressions - Key takeaways

Learn with 0 Addition and Subtraction of Rational Expressions flashcards in the free StudySmarter app

Frequently Asked Questions about Addition and Subtraction of Rational Expressions

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Addition and Subtraction of Rational Expressions

StudySmarter Editorial Team

Addition and Subtraction of Rational Expressions

The Golden Rule for Adding and Subtracting Fractions

Sum Rational Expressions with Common Denominators

Sum Rational Expressions with Different Denominators

Finding the Lowest Common Multiple of Two Polynomials

Addition and Subtraction of Rational Expressions - Key takeaways

Learn with 0 Addition and Subtraction of Rational Expressions flashcards in the free StudySmarter app

Frequently Asked Questions about Addition and Subtraction of Rational Expressions

Discover learning materials with the free StudySmarter app

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