We are asked to explain how to add and subtract rational expressions. Recall that a rational expression is a fraction whose numerator and denominator are polynomials.

Rational Expression : \frac{polynomial}{polynomial}

When adding and subtracting rational expressions we can use the same rules as when adding and subtracting numerical fractions. Therefore, we can add and subtract rational expressions only if they have the same denominator. If the expressions have different denominators, we can rewrite them using their least common denominator. Let's review these two cases!

Adding and Subtracting Rational Expressions With Like Denominators

If rational expressions have the same denominator, we add them by adding their numerators.

Sum	Add Numerators	Simplify Numerator
$\frac{1}{x ^{2}} + \frac{3}{x ^{2}}$	$\frac{1 + 3}{x ^{2}}$	$\frac{4}{x ^{2}}$
$\frac{y + 1}{2 y + 5} + \frac{2 y - 1}{2 y + 5}$	$\frac{y + 1 + 2 y - 1}{2 y + 5}$	$\frac{3 y}{2 y + 5}$
$\frac{z ^{2} + 3}{z + 1} + \frac{z - 1}{z + 1}$	$\frac{z ^{2} + 3 + z - 1}{z + 1}$	$\frac{z ^{2} + z + 2}{z + 1}$

Likewise, if rational expressions have the same denominator, we subtract them by subtracting their numerators. When the numerator which is being subtracted has more than one term, we have to put parentheses around it. Then we can use the Distributive Property to further simplify our result.

Difference	Subtract Numerators	Distributive Property	Simplify Numerator
$\frac{1}{x ^{2}} - \frac{3}{x ^{2}}$	$\frac{1 - 3}{x ^{2}}$	Does not apply	$\frac{- 2}{x ^{2}}$
$\frac{y + 1}{2 y + 5} - \frac{2 y - 1}{2 y + 5}$	$\frac{y + 1 - ( 2 y - 1 )}{2 y + 5}$	$\frac{y + 1 - 2 y + 1}{2 y + 5}$	$\frac{2 - y}{2 y + 5}$
$\frac{z ^{2} + 3}{z + 1} - \frac{z - 1}{z + 1}$	$\frac{z ^{2} + 3 - ( z - 1 )}{z + 1}$	$\frac{z ^{2} + 3 - z + 1}{z + 1}$	$\frac{z ^{2} - z + 4}{z + 1}$

Adding and Subtracting Rational Expressions With Different Denominators

If the rational expressions that are being added or subtracted have different denominators, we can take the following steps.

Find the least common denominator (LCD) of the rational expressions.
Rewrite each expression so that its denominator is equal to the LCD.
Add or subtract the rewritten expressions.

Let's see how it works on an example.

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

To find the LCD of the denominators,

x - 1

and

x^{2} - x,

we will first factor them.

x - 1 x^{2} - x = x - 1 = x \cdot (x - 1)

Knowing the factors of each denominator, we can write their LCD.

LCD = x \cdot (x - 1) = x^{2} - x

In our case, the LCD is equal to the denominator of the second expression. Therefore, we only need to rewrite the first expression. This can be done by multiplying the numerator and the denominator of

\frac{2 x}{x - 1}

by

x .

\frac{2 x}{x - 1}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot x}{b \cdot x}$

\frac{2 x \cdot x}{( x - 1 ) \cdot x}

▼

Simplify

Distr

Distribute $x$

\frac{2 x \cdot x}{x \cdot x - x}

ProdToPowTwoFac

$a \cdot a = a^{2}$

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

Now we can rewrite our sum so that the rational expressions have the same denominators.

\frac{2 x}{x - 1} + \frac{3}{x ^{2} - x} = \frac{2 x ^{2}}{x ^{2} - x} + \frac{3}{x ^{2} - x}

Finally, we can add the rational expressions.

\frac{2 x ^{2}}{x ^{2} - x} + \frac{3}{x ^{2} - x} = \frac{2 x ^{2} + 3}{x ^{2} - x}

Below there are other examples of adding and subtracting rational expressions with different denominators.

Sum or Difference	LCD	Rewritten Expressions	Result
$\frac{2}{y} + \frac{y}{y + 1}$	$y (y + 1) = y^{2} + y$	$\frac{2 y + 2}{y ^{2} + y} + \frac{y ^{2}}{y ^{2} + y}$	$\frac{y ^{2} + 2 y + 2}{y ^{2} + y}$
$\frac{6}{5 x ^{2}} - \frac{1}{2 x}$	$5 \cdot x \cdot x \cdot 2 = 10 x^{2}$	$\frac{1 2}{1 0 x ^{2}} - \frac{5 x}{1 0 x ^{2}}$	$\frac{1 2 - 5 x}{1 0 x ^{2}}$
$\frac{x ^{2}}{x ^{2} - 1} - \frac{x}{x + 1}$	$(x - 1) (x + 1) = x^{2} - 1$	$\frac{x ^{2}}{x ^{2} - 1} - \frac{x ^{2} - x}{x ^{2} - 1}$	$\frac{x}{x ^{2} - 1}$

Exercise

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