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How to Subtract Rational Expressions


The adding rational expressions page showed how to approach adding rational expressions when they appear in Math.

This page will show how to subtract rational expressions, which turns out to follow the same basic principle as adding rational expressions.




Subtracting Number Fractions Recap:

If we wanted to add or subtract a pair of fractions involving whole numbers.

We multiplied each fraction through by the denominator of the other to get a common denominator, then carrying out the addition or subtraction.

\bf{\frac{1}{2}}\bf{\frac{7}{5}}    =    \bf{\frac{1 \times 5}{2 \times 5}}\bf{\frac{7 \times 2}{5 \times 2}}    =    \bf{\frac{5}{10}}\bf{\frac{14}{10}}    =    – \bf{\frac{9}{10}}



But there is also a method we could use which is to look for what would be the lowest common denominator between the fractions.

Listing the factors of  2  and  5  would show that the lowest common denominator is  10.

Factors of  2:   2 , 4 , 6 , 8 , 10 , 12 …….

Factors of  5:   5 , 10 , 15 …….


We can then multiply each fraction by what was necessary to get the denominator to this value.

\bf{\frac{1}{2}} × \bf{\frac{5}{5}}   =   \bf{\frac{5}{10}}     ,     \bf{\frac{7}{5}} × \bf{\frac{2}{2}}   =   \bf{\frac{14}{10}}     =>     \bf{\frac{5}{10}}\bf{\frac{14}{10}}    =    – \bf{\frac{9}{10}}


This is the approach we take with subtracting rational expressions examples when we face them.




Subtracting Rational Expressions:

We can consider a sum like the following.     \frac{4}{x}\frac{5}{3x}

Similar to before with numbers, the lowest common denominator for these expressions is  3x.


As one of the fractions already has  3x  as the denominator, we only need to multiply the other fraction through.

\frac{4}{x} × \frac{3}{3}  −  \frac{5}{3x}    =    \frac{12}{3x}\frac{5}{3x}   =   \frac{12 \space {\text{–}} \space 5}{3x}    =    \frac{7}{3x}


This method also works with larger sums involving addition and subtraction of rational expressions.






How to Subtract Rational Expressions
Examples



(1.1) 

Simplify    \frac{2}{7x}\frac{1}{x}.

Solution   

\frac{2}{7x}\frac{1}{x} × \frac{7}{7}    =    \frac{2}{7x}\frac{7}{7x}    =    \frac{2 \space {\text{–}} \space 7}{7x}    =    – \frac{5}{7x}




(1.2) 

Simplify    \frac{6x \space {\text{–}} \space 4}{x \space + \space 5}\frac{2x \space + \space 7}{x \space + \space 5}.

Solution   

\frac{6x \space {\text{–}} \space 4}{x \space + \space 5}\frac{2x \space + \space 7}{x \space + \space 5}    =    \frac{(6x \space {\text{–}} \space 4) \space {\text{–}} \space (2x \space + \space 7)}{x \space + \space 5}

=    \frac{6x \space {\text{–}} \space 2x \space {\text{–}} \space 4 \space {\text{–}} \space 7}{x \space + \space 5}    =    \frac{4x \space {\text{–}} \space 11}{x \space + \space 5}




(1.3) 

Simplify    \frac{3x}{x \space {\text{–}} \space 4}\frac{5}{x \space + \space 2}.

Solution   

In this example we have to do a little bit more to get a common denominator, but the process is the same.

We can multiply the first expression by   \frac{x \space + \space 2}{x \space + \space 2},  and the second expression by  \frac{x \space {\text{–}} \space 4}{x \space {\text{–}} \space 4}.


\frac{3x}{x \space {\text{–}} \space 4} × \frac{x \space + \space 2}{x \space + \space 2}   −   \frac{5}{x \space + \space 2} × \frac{x \space {\text{–}} \space 4}{x \space {\text{–}} \space 4}

=    \frac{3x^2 \space + \space 6x}{(x \space {\text{–}} \space 4)(x \space + \space 2)}   −   \frac{5x \space {\text{–}} \space 20}{(x \space {\text{–}} \space 4)(x \space + \space 2)}

=    \frac{(3x^2 \space + \space 6x) \space\space {\text{–}} \space\space (5x \space {\text{–}} \space 20)}{(x \space {\text{–}} \space 4)(x \space + \space 2)}

=    \frac{3x^2 \space + \space 6x \space {\text{–}} \space 5x \space + \space 20}{(x \space {\text{–}} \space 4)(x \space + \space 2)}    =    \frac{3x^2 \space + \space x \space + \space 20}{(x \space {\text{–}} \space 4)(x \space + \space 2)}





Subtracting Rational Expressions Summary:


1)  Write the fractions that are present so that they share a common denominator.

2)  Write fractions as a whole fraction with this denominator.

3)  Add or subtract out the terms in the numerator on top.

4)  Reduce and simplify the fraction to the lowest and fewest terms that you can.





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