Example

Simplify[latex]\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}[/latex], and give the domain. State the result in simplest form.

Answer: 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].

[latex]\begin{array}{l}x^{2}-4=\left(x+2\right)\left(x-2\right)\\\,\,x-2=x-2\\\,\,x+2=x+2\\\,\,\text{LCM}=\left(x+2\right)\left(x-2\right)\end{array}[/latex]

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

[latex]\begin{array}{r}\frac{2{{x}^{2}}}{{{x}^{2}}-4}=\frac{2{{x}^{2}}}{(x+2)(x-2)}\\\frac{x}{x-2}\cdot \frac{x+2}{x+2}=\frac{x(x+2)}{(x+2)(x-2)}\end{array}[/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.

[latex] \displaystyle \frac{2{{x}^{2}}}{(x+2)(x-2)}+\frac{x(x+2)}{(x+2)(x-2)}[/latex]

Combine the numerators.

[latex] \begin{array}{c}\frac{2{{x}^{2}}+x(x+2)}{(x+2)(x-2)}\\\\\frac{2{{x}^{2}}+{{x}^{2}}+2x}{(x+2)(x-2)}\end{array}[/latex]

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}}+2x[/latex], this expression is in simplest form.

[latex]\frac{3{{x}^{2}}+2x}{(x+2)(x-2)}[/latex]

Answer

[latex-display] \displaystyle \frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}=\frac{3{{x}^{2}}+2x}{(x+2)(x-2)}[/latex]    [latex] \displaystyle x\ne 2,-2[/latex-display]

Example

Subtract[latex]\frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}[/latex], define the domain. State the difference in simplest form.

Answer: Find the LCD of each expression. [latex]t+1[/latex] cannot be factored any further, but [latex]{{t}^{2}}-t-2[/latex] can be. Note that t cannot be [latex]-1[/latex] or [latex]2[/latex] because the denominators would be [latex]0[/latex].

[latex]\begin{array}{c}t+1=t+1\\t^{2}-t-2=\left(t-2\right)\left(t+1\right)\end{array}[/latex]

Find the least common multiple. [latex]t+1[/latex] appears exactly once in both of the expressions, so it will appear once in the least common denominator. [latex]t–2[/latex] also appears once. This means that [latex]\left(t-2\right)\left(t+1\right)[/latex] is the least common multiple. In this case, it is easier to leave the common multiple in terms of the factors, so you will not multiply it out. Use the least common multiple for your new common denominator, it will be the LCD.

[latex]\begin{array}{c}t+1=t+1\\t^{2}-t-2=\left(t-2\right)\left(t+1\right)\\\text{LCM}:\left(t+1\right)\left(t-1\right)\end{array}[/latex]

Compare each original denominator and the new common denominator. Now rewrite the rational expressions to each have the common denominator of [latex]\left(t+1\right)\left(t–2\right)[/latex]. You need to multiply [latex]t+1[/latex] by [latex]t–2[/latex] to get the LCD, so multiply the entire rational expression by [latex] \displaystyle \frac{t-2}{t-2}[/latex]. The second expression already has a denominator of [latex]\left(t+1\right)\left(t–2\right)[/latex], so you do not need to multiply it by anything.

[latex] \begin{array}{c}\frac{2}{t+1}\cdot \frac{t-2}{t-2}=\frac{2(t-2)}{(t+1)(t-2)}\\\\\,\,\,\frac{t-2}{{{t}^{2}}-t-2}=\frac{t-2}{(t+1)(t-2)}\end{array}[/latex]

Then rewrite the subtraction problem with the common denominator.

[latex] \frac{2\left(t-2\right)}{\left(t+1\right)\left(t-2\right)}-\frac{t-2}{\left(t+1\right)\left(t-2\right)}[/latex]

Subtract the numerators and simplify. Remember that parentheses need to be included around the second [latex]\left(t–2\right)[/latex] in the numerator because the whole quantity is subtracted. Otherwise you would be subtracting just the [latex]t[/latex].

[latex] \begin{array}{c}\frac{2(t-2)-(t-2)}{(t+1)(t-2)}\\\\\frac{2t-4-t+2}{(t+1)(t-2)}\\\\\frac{t-2}{(t+1)(t-2)}\end{array}[/latex]

The numerator and denominator have a common factor of [latex]t–2[/latex], so the rational expression can be simplified.

[latex]\large\begin{array}{c}\frac{\cancel{t-2}}{(t+1)\cancel{(t-2)}}\\\\=\frac{1}{t+1}\end{array}[/latex]

Answer

[latex-display] \displaystyle \frac{2}{t+1}-\frac{t-2}{{{t}^{2}}-t-2}=\frac{1}{t+1}[/latex],   [latex]t\ne -1,2[/latex-display]