As well as simplifying rational expressions like we do with whole number fractions, we can also add and subtract rational expressions with each other.

This page will show how to approach adding rational expressions examples along with also cases of subtracting rational expressions.

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{19}{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{19}{10}}

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

We can consider a sum like the following.     \frac{5}{x} + \frac{3}{2x}

Similar to before with numbers, the lowest common denominator for these expressions is  2x.

As one of the fractions already has  2x  as the denominator, we only need to multiply the other fraction through.

\frac{5}{x} × \frac{2}{2}  +  \frac{3}{2x}    =    \frac{10}{2x} + \frac{3}{2x}   =   \frac{10 \space + \space 3}{2x}    =    \frac{13}{2x}

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

(1.1)

Simplify    \frac{4}{x} + \frac{5}{x^{\tiny{2}}}.

Solution

Here it is  x^2  that would be the lowest common denominator.

So we only need to multiply one of the fractions to then carry out the addition.

\frac{4}{x} × \frac{x}{x}  +  \frac{5}{x^{\tiny{2}}}    =    \frac{4x}{x^{\tiny{2}}} + \frac{5}{x^{\tiny{2}}}    =    \frac{4x \space + \space 5}{x^{\tiny{2}}}

(1.2)

Simplify    \frac{4}{2(x \space {\text{--}} \space 3)} + \frac{x \space + \space 1}{x^{\tiny{2}} \space + \space x \space {\text{--}} \space 12}.

Solution

With a situation like this, the best first step is to try to factor the denominator on the fraction with the quadratic present.

\frac{4}{2(x \space {\text{--}} \space 3)} + \frac{x \space + \space 1}{(x \space {\text{--}} \space 3)(x \space + \space 4)}

Things are now looking a bit clearer.
We can put together a common denominator of  2(x \space {\text{--}} \space 3)(x \space + \space 4)

This can be achieved by multiply the first expression by  \frac{x \space + \space 4}{(x \space + \space 4)},  and the second expression by  \frac{2}{2}.

\frac{4}{2(x \space {\text{--}} \space 3)} × \frac{(x \space + \space 4)}{(x \space + \space 4)}   +   \frac{x \space + \space 1}{(x \space {\text{--}} \space 3)(x \space + \space 4)} × \frac{2}{2}

=   \frac{4(x \space + \space 4)}{2(x \space {\text{--}} \space 3)(x \space + \space 4)} + \frac{2x \space + \space 2}{2(x \space {\text{--}} \space 3)(x \space + \space 4)}

=   \frac{\space 4x \space + \space 16 \space + \space 2x + 2 \space}{2(x \space {\text{--}} \space 3)(x \space + \space 4)}   =   \frac{6x \space + \space 18}{2(x \space {\text{--}} \space 3)(x \space + \space 4)}

=   \frac{6(x \space + \space 3)}{2(x \space {\text{--}} \space 3)(x \space + \space 4)}   =   \frac{3(x \space + \space 3)}{(x \space {\text{--}} \space 3)(x \space + \space 4)}

We have now factored and simplified as far as possible.

(1.3)

Simplify    \frac{3}{a} + \frac{2}{a^{\tiny{2}}} + \frac{1}{4a}.

Solution

Having three expressions to add together doesn’t change our approach, we still look to carry out multiplication that will give us a common denominator.

From looking at the expressions we can see that the lowest common denominator we can obtain would be  4a^2
.

\frac{3}{a} × \frac{4a}{4a}   +   \frac{2}{a^{\tiny{2}}} × \frac{4}{4}   +   \frac{1}{4a} × \frac{a}{a}

=    \frac{12a}{4a^{\tiny{2}}}  +  \frac{8}{4a^{\tiny{2}}}  +   \frac{a}{4a^{\tiny{2}}}

=    \frac{12a \space + \space 8 \space + \space a}{4a^{\tiny{2}}}    =    \frac{13a \space + \space 8}{4a^{\tiny{2}}}

We have now factored and simplified as far as possible.

1)  Rewrite all the fractions present so that they share a common denominator.

2)  Write as one whole fraction with this denominator.

3)  Add or subtract out the numerator on top.

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

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