Rational Expressions
Marcus can wash his family's cars alone in $x$ hours. His sister Aaliyah needs $x + 2$ hours. How much of the job do they finish in one hour if they work together? The answer is $\dfrac{1}{x} + \dfrac{1}{x+2}$, a sum with a variable in the denominator. Expressions like this are called rational expressions. You work with them exactly like numerical fractions, only with polynomials on top and bottom.
What is a rational expression?
To evaluate a rational expression, substitute the number for the variable and compute the top and the bottom separately.
Excluded values
A fraction is not defined when its denominator is $0$. So in $\dfrac{2x-1}{x^2-4}$ we must have $x^2 - 4 \neq 0$. Factor: $(x+2)(x-2) = 0$ when $x = -2$ or $x = 2$. Those are the excluded values.
Look at the graph of $\dfrac{x-3}{x+1}$ near $x = -1$: the expression is undefined there, so the graph has a break.
Evaluate the algebraic fraction $\frac{x^2 - 3x + 2}{x^2 + 1}$ for $x = -2$.
Evaluate the algebraic fraction $\frac{x^2 - 3x + 2}{x^2 + 1}$ for $x = -2$.
Simplifying with a common factor
To simplify a rational expression, factor the numerator and the denominator, then cancel factors that appear in both. It is the same idea as $\dfrac{12}{18} = \dfrac{2 \cdot 2 \cdot 3}{2 \cdot 3 \cdot 3} = \dfrac{2}{3}$. The expression is in simplest form when no common factor is left.
The first tool is the greatest common factor (GCF): factor it out of every term, top and bottom.
Simplify the following algebraic fraction by factoring out the common factor in the numerator and denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$
Simplify the following algebraic fraction by factoring out the common factor in the numerator and denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$
The total cost in dollars of manufacturing $x$ bicycles is given by the polynomial $C(x) = 3x^2 + 150x$. The total revenue in dollars from selling these $x$ bicycles is…
The total cost in dollars of manufacturing $x$ bicycles is given by the polynomial $C(x) = 3x^2 + 150x$. The total revenue in dollars from selling these $x$ bicycles is expressed by the polynomial $I(x) = 5x^2 + 200x$. Write the simplified algebraic fraction that represents the ratio of the revenue to the costs.
Simplifying by factoring polynomials
When there is no common factor to pull out, use your whole factoring toolbox.
| Pattern | Factored form |
|---|---|
| Difference of squares | $a^2 - b^2 = (a+b)(a-b)$ |
| Perfect square trinomial | $a^2 \pm 2ab + b^2 = (a \pm b)^2$ |
| Trinomial $x^2 + bx + c$ | $(x + p)(x + q)$ with $pq = c$ and $p + q = b$ |
| Four terms | factor by grouping |
Simplify the following algebraic fraction by applying algebraic identities: $\frac{x^2 - 25}{x^2 - 10x + 25}$
Simplify the following algebraic fraction by applying algebraic identities: $\frac{x^2 - 25}{x^2 - 10x + 25}$
Simplify the following algebraic fraction by first factoring the numerator and the denominator: $\frac{x^3 - 4x^2 + 4x}{x^3 - 2x^2 - 4x + 8}$
Simplify the following algebraic fraction by first factoring the numerator and the denominator: $\frac{x^3 - 4x^2 + 4x}{x^3 - 2x^2 - 4x + 8}$
Equivalent rational expressions
Two fractions $\dfrac{A}{B}$ and $\dfrac{C}{D}$ are equivalent when $A \cdot D = B \cdot C$. This is the cross-multiplying test. It works for polynomials, too.
The same idea finds a missing polynomial. If $\dfrac{x^2-4}{P(x)} = \dfrac{x-2}{x^2+3x}$, then $P(x)$ must be the denominator multiplied by whatever turned $x - 2$ into $x^2 - 4$.
Check, by cross-multiplying, if the algebraic fractions $\frac{x-3}{x+1}$ and $\frac{x^2-9}{x^2+4x+3}$ are equivalent.
Check, by cross-multiplying, if the algebraic fractions $\frac{x-3}{x+1}$ and $\frac{x^2-9}{x^2+4x+3}$ are equivalent.
Find the polynomial $P(x)$ that makes the following equality between algebraic fractions true: $\frac{x^2-9}{P(x)} = \frac{x-3}{x^2+2x}$
Find the polynomial $P(x)$ that makes the following equality between algebraic fractions true: $\frac{x^2-9}{P(x)} = \frac{x-3}{x^2+2x}$
Multiplying and dividing
Multiply numerator times numerator and denominator times denominator. To divide, multiply by the reciprocal of the second fraction. Factor first so you can cancel along the way.
Multiply the following algebraic fractions and write the result in simplest form: $\frac{x}{x+2} \cdot \frac{x^2-4}{x^2}$
Multiply the following algebraic fractions and write the result in simplest form: $\frac{x}{x+2} \cdot \frac{x^2-4}{x^2}$
Divide the following algebraic fractions and simplify: $\frac{2x+2}{x-1} : \frac{4x+4}{x^2-1}$
Divide the following algebraic fractions and simplify: $\frac{2x+2}{x-1} : \frac{4x+4}{x^2-1}$
Adding and subtracting with the same denominator
When the denominators match, keep the denominator and add or subtract the numerators. Then simplify.
Add the following algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$
Add the following algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$
Perform the following operation and simplify the result: $\frac{x^2+5x}{x+3} - \frac{x^2-9}{x+3}$
Perform the following operation and simplify the result: $\frac{x^2+5x}{x+3} - \frac{x^2-9}{x+3}$
Adding and subtracting with different denominators
You need a least common denominator (LCD). Factor each denominator, then take every factor the greatest number of times it appears in any one denominator.
The steps are: factor the denominators, find the LCD, rewrite each fraction with the LCD, combine the numerators, simplify.
Perform the following operations by finding a common denominator and simplify the result: $\frac{2}{x-1} + \frac{x}{x+1} - \frac{2x}{x^2-1}$
Perform the following operations by finding a common denominator and simplify the result: $\frac{2}{x-1} + \frac{x}{x+1} - \frac{2x}{x^2-1}$
Subtract the following algebraic fractions and simplify fully: $\frac{x^2+x}{x^2-1} - \frac{x}{x-1}$
Subtract the following algebraic fractions and simplify fully: $\frac{x^2+x}{x^2-1} - \frac{x}{x-1}$
Mixed operations and complex fractions
Follow the order of operations: parentheses first, then multiplication and division from left to right.
A complex fraction has fractions in its numerator, its denominator, or both. Two ways to simplify: do the small fractions first, or multiply the top and bottom by the LCD of all the small fractions.
Perform the following combined operation with algebraic fractions and simplify the result: $\left( \frac{1}{x} - \frac{2}{x+2} \right) : \frac{x-2}{x^2+2x}$
Perform the following combined operation with algebraic fractions and simplify the result: $\left( \frac{1}{x} - \frac{2}{x+2} \right) : \frac{x-2}{x^2+2x}$
Simplify the following algebraic expression in the form of a complex fraction: $1 + \frac{1}{1 + \frac{1}{x-1}}$
Simplify the following algebraic expression in the form of a complex fraction: $1 + \frac{1}{1 + \frac{1}{x-1}}$
Simplify the following expression fully: $\frac{x^2-1}{x^2+2x+1} \cdot \left( \frac{2x}{x-1} - \frac{x+1}{x} \right)$
Simplify the following expression fully: $\frac{x^2-1}{x^2+2x+1} \cdot \left( \frac{2x}{x-1} - \frac{x+1}{x} \right)$
Special cases: finding a parameter and sums of cubes
Making a fraction simplifiable
If $\dfrac{N(x)}{x - 3}$ can be simplified, then $x - 3$ is a factor of $N(x)$. A product that has the factor $(x-3)$ equals $0$ when $x = 3$. So substitute $x = 3$ into the numerator and set it equal to $0$.
Sums and differences of cubes
Two more factoring patterns help with many problems:
Determine the value of the parameter $k$ so that the algebraic fraction $\frac{x^2 - kx + 10}{x-5}$ can be simplified (that is, the numerator is divisible by the…
Determine the value of the parameter $k$ so that the algebraic fraction $\frac{x^2 - kx + 10}{x-5}$ can be simplified (that is, the numerator is divisible by the denominator). Once $k$ is found, write the simplified expression.
Show, by expanding and simplifying the algebraic fractions, that the following expression is equal to $2ab$: $\frac{a^3-b^3}{a-b} - \frac{a^3+b^3}{a+b}$
Show, by expanding and simplifying the algebraic fractions, that the following expression is equal to $2ab$: $\frac{a^3-b^3}{a-b} - \frac{a^3+b^3}{a+b}$
Modeling rates and work
Two formulas turn word problems into rational expressions:
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Work: if a job takes $t$ hours, you finish $\dfrac{1}{t}$ of it per hour. Rates add up when people work together.
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Distance: $\text{time} = \dfrac{\text{distance}}{\text{rate}}$.
Back to Marcus and Aaliyah. In one hour Marcus washes $\dfrac{1}{x}$ of the job and Aaliyah washes $\dfrac{1}{x+2}$ of it. The LCD is $x(x+2)$:
$$\dfrac{1}{x} + \dfrac{1}{x+2} = \dfrac{x+2}{x(x+2)} + \dfrac{x}{x(x+2)} = \dfrac{2x+2}{x(x+2)}$$For example, if $x = 2$ (Marcus takes 2 hours and Aaliyah 4), the formula gives $\dfrac{6}{8} = \dfrac{3}{4}$ of the job per hour. Check: $\dfrac{1}{2} + \dfrac{1}{4} = \dfrac{3}{4}$.
A hot water faucet takes $x$ hours to fill a bathtub, while the cold water faucet takes $x+3$ hours. Write a simplified algebraic fraction that represents the portion of…
A hot water faucet takes $x$ hours to fill a bathtub, while the cold water faucet takes $x+3$ hours. Write a simplified algebraic fraction that represents the portion of the bathtub filled in one hour if both faucets are turned on at the same time.
A train travels a distance of $120$ miles at a constant speed of $v$ mph. On the return trip, due to rain, its speed decreases by $10$ mph. Write the algebraic fraction…
A train travels a distance of $120$ miles at a constant speed of $v$ mph. On the return trip, due to rain, its speed decreases by $10$ mph. Write the algebraic fraction that represents the total time of the round trip in terms of $v$, combining it into a single fraction.
An athlete runs on a circular track. During the first half of the distance, they run at a speed of $v$ ft/s, and during the second half, they run at a speed of $v-2$…
An athlete runs on a circular track. During the first half of the distance, they run at a speed of $v$ ft/s, and during the second half, they run at a speed of $v-2$ ft/s. Knowing that the average speed for the whole trip is calculated as total distance divided by total time, show using algebraic fractions that the average speed is $\frac{2v(v-2)}{2v-2}$ ft/s.
Modeling with area, volume and cost
Write each quantity as a polynomial, make a fraction of the right pair, and simplify.
For an open box, cut squares from the corners of a sheet and fold up the sides.
Cost problems work the same way: price per unit = total cost ÷ total amount.
The base of a rectangular solar panel measures $x+5$ feet and its height measures $x-5$ feet. The side length of a square solar panel measures $x$ feet. Write the…
The base of a rectangular solar panel measures $x+5$ feet and its height measures $x-5$ feet. The side length of a square solar panel measures $x$ feet. Write the algebraic fraction that represents the ratio of the area of the rectangular panel to the area of the square panel. Find the value of this fraction if $x = 10$ feet.
An open box is to be made from a rectangular sheet of cardboard with dimensions $x+6$ and $x+4$ inches, by cutting out squares with side length $2$ in. from the four…
An open box is to be made from a rectangular sheet of cardboard with dimensions $x+6$ and $x+4$ inches, by cutting out squares with side length $2$ in. from the four corners and folding up the flaps. Write the simplified algebraic fraction that represents the ratio of the volume of the box to the area of its base.
In a chemistry laboratory, $x$ gallons of solution A, which costs $\$\frac{20}{x}$ per gallon, are mixed with $x+4$ gallons of solution B, which costs $\$\frac{30}{x+4}$…
In a chemistry laboratory, $x$ gallons of solution A, which costs $\$\frac{20}{x}$ per gallon, are mixed with $x+4$ gallons of solution B, which costs $\$\frac{30}{x+4}$ per gallon. Write the algebraic fraction that represents the price per gallon of the resulting solution, simplifying the expression fully.