Rational expression
A rational expression is an algebraic expression of the form $\frac{P}{Q},$ where $P$ and $Q$ are simpler expressions (usually polynomials), and the denominator $Q$ is not zero.
A rational expression is an algebraic expression of the form $\frac{P}{Q},$ where $P$ and $Q$ are simpler expressions (usually polynomials), and the denominator $Q$ is not zero.
Examples
$\frac{1}{x-1}$ \t \gap[40] \t is rational with $\color{slateblue}{P = 1,\ \ Q = x-1}$
\\ \t
\\ $\frac{x^2+3x+1}{x^2+3}$ \t \t is rational with $\color{slateblue}{P = x^2+3x+1,\ \ Q = x^2+3}$
\\ \t
Note
As with numbers, we can think of expressions with no denominator (like whole numbers or polynomials) as rational expressions by dividing them by 1:
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$x$ can be thought of as $\frac{x}{1},$ which is rational with $\color{slateblue}{P = x},$ and $\color{slateblue}{Q = 1}.$
$1$ can be thought of as $\frac{1}{1},$ which is rational with $\color{slateblue}{P = 1},$ and $\color{slateblue}{Q = 1}.$
$x^2y-2xy^2+1$ can be thought of as $\frac{x^2y-2xy^2+1}{1},$ which is rational with $\color{slateblue}{P = x^2y-2xy^2+1},$ and $\color{slateblue}{Q = 1}.$
Algebra of rational expressions
The rules for manipulating rational expressions are the same as the rules for manipulating fractions. Let's look at these rules one-by-one:
The cancellation rule: Simplifying a rational expression
Cancellation rule
If $R$ is any nonzero expression that is a factor of both the numerator and denominator, then you can cancel it to simplify the rational expression:
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$\frac{P\color{indianred}{R}}{Q\color{indianred}{R}} = \frac{P}{Q} \qquad \quad \ \ $ Cancel the R.
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$\frac{P+\color{indianred}{R}}{Q+\color{indianred}{R}} \neq \frac{P}{R} \qquad$ You cannot cancel a summand.
Examples
\\ !r! $\frac{(x^2+3x+1)\color{indianred}{(x-1)}}{(x^2+3)\color{indianred}{(x-1)}}$ \t $ = \frac{x^2+3x+1}{x^2+3}$ \t Cancel the factor $\color{#6968d0}{x-1}$.
\\ \t
\\ !r! $\frac{\color{indianred}{x^2}}{\color{indianred}{x^2}(x-1)}$ \t $= \frac{1}{x-1}$ \t Cancel the factor $\color{#6968d0}{x^2}$.
\\ \t
\\ $\frac{\color{indianred}{(x^2-1)}(x^2y-2xy^2+1)}{\color{indianred}{(x^2-1)}}$ \t $= \frac{x^2y-2xy^2+1}{1}$ \t Cancel the factor $\color{#6968d0}{x^2-1}$.
\\ \t $= x^2y-2xy^2+1$
\\ \t
\\ !r!$\frac{6x^2}{10x^4}$ \t $= \frac{3 \cdot \color{indianred}{2 \cdot x \cdot x}}{5 \cdot \color{indianred}{2 \cdot x \cdot x} \cdot x \cdot x}$ \t Factor.
\\ \t $= \frac{3}{5x^2}$ \t Cancel the factor $\color{#6968d0}{2x^2}$.
\\ \t
\\ !r!$\frac{x^3+x}{x^2+x}$ \t $= \frac{\color{indianred}{x}(x^2+1)}{\color{indianred}{x}(x+1)}$ \t Factor.
\\ \t $= \frac{x^2+1}{x+1}$ \t Cancel the factor $\color{#6968d0}{x}$.
\\ \t
\\ !r!$\frac{x^2+3x+2}{x+1}$ \t $= \frac{\color{indianred}{(x+1)}(x+2)}{\color{indianred}{x+1}}$ \t Factor.
\\ \t $= x+2$ \t Cancel the factor $\color{#6968d0}{x+1}$.
Some for you to do
Multiplying and dividing rational expressions
Multiplying rational expressions
As is the case with ordinary fractions, we multiply two rational expressions by simply multiplying their numerators and denominators:
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$\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} \times \frac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \frac{\color{#026fc1}{P}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $ Product of numerators divided by product of denominators
Examples
\\ !r! $\frac{\color{#026fc1}{x+1}}{\color{#026fc1}{x}} \times \frac{\color{#c1026f}{x-1}}{\color{#c1026f}{2x+1}} $ \t $= \frac{\color{#026fc1}{(x+1)}\color{#c1026f}{(x-1)}}{\color{#026fc1}{x}\color{#c1026f}{(2x+1)}}$ \t Multiply top and bottom.
\\ \t $= \frac{x^2-1}{2x^2+1}$ \t Calculate the products.
\\ \t
\\ !r! $\color{#026fc1}{2} \times \frac{\color{#c1026f}{4x}}{\color{#c1026f}{x-1}} $ \t $= \frac{\color{#026fc1}{2}}{\color{#026fc1}{1}} \times \frac{\color{#c1026f}{4x}}{\color{#c1026f}{x-1}} $ \t Convert to a rational expression.
\\ \t $= \frac{\color{#026fc1}{2}\color{#c1026f}{(4x)}}{\color{#026fc1}{(1)}\color{#c1026f}{(x-1)}}$ \t Multiply top and bottom.
\\ \t $= \frac{8x}{x-1}$ \t Calculate the products.
\\ \t
\\ !r! $\frac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}} \times \color{#c1026f}{(x^2+1)} $ \t $= \frac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}} \times \frac{\color{#c1026f}{x^2+1}}{\color{#c1026f}{1}} $ \t Convert to a rational expression.
\\ \t $= \frac{\color{#026fc1}{4x}\color{#c1026f}{(x^2+1)}}{\color{#026fc1}{(x-1)}\color{#c1026f}{(1)}}$ \t Multiply top and bottom.
\\ \t $= \frac{4x^3+4x}{x-1}$ \t Calculate the products.
\\ \t
\\ !r! $\frac{\color{#026fc1}{x-4}}{\color{#026fc1}{6x}} \times \frac{\color{#c1026f}{4x^3}}{\color{#c1026f}{2x+1}} $ \t $= \frac{\color{#026fc1}{(x-4)}\color{#c1026f}{4x^3}}{\color{#026fc1}{6x}\color{#c1026f}{(2x+1)}}$ \t Multiply top and bottom.
\\ \t $=\frac{(x-4)2x}{3(2x+1)}$ \t Simplify: Cancel $\color{#6968d0}{2x}$.
\\ \t $= \frac{2x^2-8x}{6x+3}$ \t Calculate the products.
\\ \t
\\ !r! $\frac{\color{#026fc1}{x-1}}{\color{#026fc1}{x}} \times \frac{\color{#c1026f}{2x^2-x}}{\color{#c1026f}{x^2-2x-1}} $ \t $= \frac{\color{#026fc1}{(x-1)}\color{#c1026f}{(2x^2-x)}}{\color{#026fc1}{x}\color{#c1026f}{(x^2-2x-1)}}$ \t Multiply top and bottom.
\\ \t $=\frac{(x-1)(x)(2x-1)}{x(x-1)(x-1)}$ \t Factor.
\\ \t $=\frac{2x-1}{x-1}$ \t Simplify: Cancel the $\color{#6968d0}{x}$ %and $\color{#6968d0}{(x-1)}$.
Some for you to do
Dividing rational expressions
As with ordinary fractions, division by a rational expression means multiplication by its reciprocal:
$\frac{\left(\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}\right)}{\left(\frac{\color{#c1026f}{R}}{\color{#c1026f}{S}}\right)}$ \t $= \frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} \times \frac{\color{#c1026f}{S}}{\color{#c1026f}{R}} \qquad \quad \ \ $ \t Flip the denominator and multiply.
\\ \t $= \frac{\color{#026fc1}{P}\color{#c1026f}{S}}{\color{#026fc1}{Q}\color{#c1026f}{R}} $ \t Calculate the product.
Note
As with products, before actually calculating the products on the top and bottom, you should first simplify by factoring and cancelling if possible.
Suggested video for this topic: %22
Examples
\\ !r! $\frac{\left(\frac{\color{#026fc1}{x+1}}{\color{#026fc1}{x}}\right)}{\left(\frac{\color{#c1026f}{x-1}}{\color{#c1026f}{2x+1}}\right)} $ \t $=\frac{\color{#026fc1}{x+1}}{\color{#026fc1}{x}} \times \frac{\color{#c1026f}{2x+1}}{\color{#c1026f}{x-1}}$ \t Flip the denominator and multiply.
\\ \t $= \frac{\color{#026fc1}{(x+1)}\color{#c1026f}{(2x+1)}}{\color{#026fc1}{x}\color{#c1026f}{(x-1)}}$ \t Calculate the product.
\\ \t $= \frac{2x^2+3x+1}{x^2-x}$
\\ \t
\\ !r! $\frac{1}{\left(\frac{\color{#c1026f}{2x-1}}{\color{#c1026f}{x^3}}\right)} $ \t $= 1 \times \frac{\color{#c1026f}{x^3}}{\color{#c1026f}{2x-1}}$ \t Flip the denominator and multiply.
\\ \t $= \frac{x^3}{2x-1}$
\\ \t
\\ !r! $\frac{\left(\frac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}}\right)}{\color{#c1026f}{x^2+1}} $ \t $= \frac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}} \times \frac{\color{#c1026f}{1}}{\color{#c1026f}{x^2+1}} $ \t Flip the denominator and multiply.
\\ \t $= \frac{\color{#026fc1}{4x}\color{#c1026f}{(1)}}{\color{#026fc1}{(x-1)}\color{#c1026f}{(x^2+1)}}$ \t Multiply top and bottom.
\\ \t $= \frac{4x}{x^3-x^2+x+1}$ \t Calculate the products.
\\ \t
\\ !r! $\frac{\color{#c1026f}{x^2+1}}{\left(\frac{\color{#026fc1}{4x}}{\color{#026fc1}{x-1}}\right)} $ \t $= \color{#c1026f}{(x^2+1)} \times \frac{\color{#026fc1}{x-1}}{\color{#026fc1}{4x}}$ \t Flip the denominator and multiply.
\\ \t $= \frac{\color{#026fc1}{(x^2+1)}\color{#c1026f}{(x-1)}}{\color{#026fc1}{(1)}\color{#c1026f}{(4x)}}$ \t Multiply top and bottom.
\\ \t $= \frac{x^3-x^2+x+1}{4x}$ \t Calculate the products.
\\ \t
Some for you to do
Adding and subtracting rational expressions
Like the other rules we have seen, the rules for addition and subtraction of rational expressions are the same as for ordinary fractions. We start with the case in which the expressions we are adding or subtracting have the same denominator.
Addition and subtraction with common denominator:
1. As with ordinary fractions, this formula works only when the two expressions have the same denominator.
2. When adding or subtracting expressions with the same denominator, do not factor or cancel before starting as you would with products or quotients; just leave them as they are until after doing the addition or subtractions. Suggested video for this topic: %23
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$\frac{\color{#026fc1}{P}}{\color{#c1026f}{Q}} + \frac{\color{#026fc1}{R}}{\color{#c1026f}{Q}} = \frac{\color{#026fc1}{P} + \color{#026fc1}{R}}{\color{#c1026f}{Q}} \qquad \quad \ \ $ Sum of numerators divided by common denominator
$\frac{\color{#026fc1}{P}}{\color{#c1026f}{Q}} - \frac{\color{#026fc1}{R}}{\color{#c1026f}{Q}} = \frac{\color{#026fc1}{P} - \color{#026fc1}{R}}{\color{#c1026f}{Q}} \qquad \quad \ \ $ Difference of numerators divided by common denominator
1. As with ordinary fractions, this formula works only when the two expressions have the same denominator.
2. When adding or subtracting expressions with the same denominator, do not factor or cancel before starting as you would with products or quotients; just leave them as they are until after doing the addition or subtractions. Suggested video for this topic: %23
Examples
\\ !r! $\frac{\color{#026fc1}{y}}{\color{#c1026f}{xy+1}} + \frac{\color{#026fc1}{x-1}}{\color{#c1026f}{xy+1}} $ \t $= \frac{\color{#026fc1}{y + x - 1}}{\color{#c1026f}{xy+1}}$ \t Add the numerators.
\\ \t
\\ !r! $\frac{\color{#026fc1}{x^2+1}}{\color{#c1026f}{x-1}} - \frac{\color{#026fc1}{2x}}{\color{#c1026f}{x-1}} $ \t $= \frac{\color{#026fc1}{x^2-2x+1}}{\color{#c1026f}{x-1}}$ \t Subtract the numerators.
\\ \t $= \frac{(x-1)^2}{x-1}$ \t Factor.
\\ \t $= x-1$ \t Simplify: Cancel the $\color{#6968d0}{(x-1)}$.
\\ \t
Some for you to do
Addition and subtraction: General case:
1. This formula works for ordinary fractions as well, and also when the two expressions have the same denominator (although cancellation is necessary to simplify the answer in that case).
2. When adding or subtracting expressions with different denominators, it helps to factor and/or cancel before starting as you would with products or quotients. This makes it easier to simplify the final answer.
3. If the denominators are the same, it is better to use the rule for addition and subtraction with common denominator; otherwise you will need to do additional work to simplify the answer. Suggested video for this topic: %24
$\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} + \frac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \frac{\color{#026fc1}{P}\color{#c1026f}{S} + \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $ | Cross multiply to get the numerator: | $\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\frac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$ |
Multiply across to get the denominator: | $\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\frac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$ | |
$\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} - \frac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \frac{\color{#026fc1}{P}\color{#c1026f}{S} - \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $ | Cross multiply to get the numerator: | $\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\frac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$ |
Multiply across to get the denominator: | $\frac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\frac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$ |
1. This formula works for ordinary fractions as well, and also when the two expressions have the same denominator (although cancellation is necessary to simplify the answer in that case).
2. When adding or subtracting expressions with different denominators, it helps to factor and/or cancel before starting as you would with products or quotients. This makes it easier to simplify the final answer.
3. If the denominators are the same, it is better to use the rule for addition and subtraction with common denominator; otherwise you will need to do additional work to simplify the answer. Suggested video for this topic: %24
Examples
\\ !r! $\frac{\color{#026fc1}{3}}{\color{#026fc1}{2x+1}} + \frac{\color{#c1026f}{4}}{\color{#c1026f}{x-5}} $ \t $= \frac{\color{#026fc1}{3}\color{#c1026f}{(x-5)} + \color{#026fc1}{(2x+1)}\color{#c1026f}{4}}{\color{#026fc1}{(2x+1)}\color{#c1026f}{(x-5)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t $= \frac{11x - 9}{(2x+1)(x-5)}$
\t
Calculate the numerator.
\\
\\ !r! $\frac{\color{#026fc1}{2x}}{\color{#026fc1}{y-1}} - \frac{\color{#c1026f}{y+1}}{\color{#c1026f}{x}} $ \t $= \frac{\color{#026fc1}{2x}\color{#c1026f}{(x)} - \color{#026fc1}{(y-1)}\color{#c1026f}{(y+1)}}{\color{#026fc1}{(y-1)}\color{#c1026f}{(x)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t $= \frac{2x^2-y^2+1}{(2x+1)(x-5)}$
\t
Calculate the numerator.
\\
\\ !r! $\frac{\color{#026fc1}{5}}{\color{#026fc1}{2x}} - \frac{\color{#c1026f}{3}}{\color{#c1026f}{2(x+5)}} $ \t $= \frac{\color{#026fc1}{5}\cdot \color{#c1026f}{2(x+5)} - \color{#026fc1}{(2x)}\color{#c1026f}{3}}{\color{#026fc1}{2x}\cdot \color{#c1026f}{2(x+5)}}$ \t
Cross multiply for the numerator and multiply across for the denominator.
\\ \t $= \frac{4x + 50}{4x(x+5)}$
\t
Calculate the numerator.
\\ \t $= \frac{2(2x + 25)}{4x(x+5)}$
\t
Factor the numerator.
\\ \t $= \frac{2x + 25}{2x(x+5)}$
\t
Simplify: Cancel the 2.
Some for you to do
Combining the rules
One last quiz in which you will need to combine the above rules:
Now try some of the exercises in Section 0.4 of or , or move ahead to the next tutorial by pressing on the sidebar.
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