Adding and Subtractracting Rational Expressions Part II

Learning Outcomes

Add and subtract rational expressions that share no common factors

Add and subtract more than two rational expressions

So far all the rational expressions you’ve added and subtracted have shared some factors. What happens when they don’t have factors in common?

No Common Factors

Add and Subtract Rational Expressions with No Common Factor

In the next example, we show how to find a common denominator when there are no common factors in the expressions.

Example

Subtract [latex] \displaystyle \frac{3y}{2y-1}-\frac{4}{y-5}[/latex], and give the domain.

State the difference in simplest form.

Show Solution

Neither [latex]2y–1[/latex] nor [latex]y–5[/latex] can be factored. Because theyhave no common factors, the least common multiple, which will become the least common denominator, is the product of these denominators.

Multiply each expression by the equivalent of [latex]1[/latex] that will give it the common denominator.

Then rewrite the subtraction problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.

In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.

Try It

Add and Subtract More Than Two Rational Expressions

You may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?

In the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms.

Example

Simplify[latex]\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}-\frac{1}{x+2}[/latex], and give the domain.

State the result in simplest form.

Show Solution

Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization. Remember that x cannot be [latex]2[/latex] or [latex]-2[/latex] because the denominators would be [latex]0[/latex].

[latex]\left(x+2\right)[/latex] appears a maximum of one time, as does [latex]\left(x–2\right)[/latex]. This means the LCM is [latex]\left(x+2\right)\left(x–2\right)[/latex].

Rewrite the original problem with the common denominator. It makes sense to keep the denominator in factored form in order to check for common factors.

Check for simplest form. Since neither [latex]\left(x+2\right)[/latex] nor [latex]\left(x-2\right)[/latex] is a factor of [latex]3{{x}^{2}}+x+2[/latex], this expression is in simplest form.

In the video that follows we present an example of subtracting [latex]3[/latex] rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is [latex]1[/latex].

Example

Simplify[latex]\frac{{{y}^{2}}}{3y}-\frac{2}{x}-\frac{15}{9}[/latex], and give the domain.

State the result in simplest form.

Show Solution

Find the least common multiple by factoring each denominator. Multiply each factor the maximum number of times it appears in a single factorization.

In this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.

Add and Subtract

Try It

Summary

The methods shown here will help you when you are solving rational equations later on. To add and subtract rational expressions that share common factors, you first identify which factors are missing from each expression, and build the LCD with them. To add and subtract rational expressions with no common factors, the LCD will be the product of all the factors of the denominators.

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Image: No common factors.. Provided by: Lumen Learning. License: CC BY: Attribution

Screenshot: Add and subtract. Provided by: Lumen Learning. License: CC BY: Attribution

Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution

Subtract Rational Expressions with UnLike Denominators - 3 Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/c-8xQyU0ch0. License: CC BY: Attribution

Add and Subtract Rational Expressions with UnLike Denominators - 3 Expressions. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/43xPStLm39A. License: CC BY: Attribution