Rational expressions
61 flashcards covering Rational expressions for the SAT Math section.
Rational expressions are fractions where the numerator and denominator are polynomials—basically, they're like regular fractions but with variables instead of just numbers. For example, something like (x^2 + 1)/(x - 2) is a rational expression. They represent ratios of algebraic expressions and are used to model real-world situations, such as rates or proportions. Understanding them helps you simplify complex equations and solve problems involving division of polynomials.
On the SAT Math section, rational expressions appear in questions about simplifying, adding, subtracting, multiplying, or dividing them, often within algebra or word problems. Common traps include forgetting to exclude values that make the denominator zero, which can lead to undefined answers, or mishandling complex fractions. Focus on mastering simplification techniques, like factoring and canceling common terms, and practice identifying restrictions on variables to avoid errors.
Remember to always check for domain restrictions before simplifying.
Terms (61)
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Rational expression
A fraction where both the numerator and the denominator are polynomials, and the denominator is not zero.
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Simplifying a rational expression
The process of factoring the numerator and denominator and then canceling any common factors, as long as the denominator is not zero after simplification.
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Numerator of a rational expression
The polynomial expression in the top part of a rational expression, which can be simplified by factoring if it shares factors with the denominator.
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Denominator of a rational expression
The polynomial expression in the bottom part of a rational expression, which must never be zero, as that would make the expression undefined.
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Factoring in rational expressions
A key step in simplifying rational expressions, where polynomials in the numerator and denominator are factored to identify and cancel common factors.
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Common factors in rational expressions
Polynomial factors that appear in both the numerator and denominator, which can be canceled to simplify the expression, provided they do not make the denominator zero.
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Least common denominator (LCD)
The simplest polynomial that is a multiple of all denominators in an expression, used when adding or subtracting rational expressions with unlike denominators.
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Adding rational expressions
The process of finding a common denominator, rewriting each fraction with that denominator, and then adding the numerators while keeping the denominator the same.
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Subtracting rational expressions
Similar to adding, but involves subtracting the numerators after rewriting with a common denominator, ensuring the result is simplified and undefined values are noted.
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Multiplying rational expressions
Multiply the numerators together and the denominators together, then simplify by factoring and canceling common factors before performing the multiplication if possible.
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Dividing rational expressions
Multiply the first rational expression by the reciprocal of the second, then simplify by factoring and canceling common factors as needed.
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Reciprocal of a rational expression
The rational expression obtained by swapping the numerator and denominator of the original, used in division of rational expressions.
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Complex fraction
A fraction where the numerator, denominator, or both are themselves fractions, which can be simplified by multiplying the top and bottom by the least common denominator of all fractions involved.
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Simplifying a complex fraction
Multiply the numerator and denominator of the complex fraction by the least common denominator of all the fractions within it to eliminate the inner fractions and then simplify.
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Rational equation
An equation containing one or more rational expressions, typically solved by multiplying both sides by the least common denominator to eliminate fractions, while checking for extraneous solutions.
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Solving a rational equation
Multiply both sides by the least common denominator to clear fractions, solve the resulting equation, and verify solutions are not values that make any original denominator zero.
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Extraneous solution
A solution that arises during the solving process of a rational equation but does not satisfy the original equation, often because it makes a denominator zero.
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Domain of a rational expression
The set of all real numbers for which the expression is defined, excluding any values that make the denominator zero.
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Values that make denominator zero
Specific numbers that, when substituted into the denominator of a rational expression, result in zero, making the expression undefined and restricting the domain.
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Vertical asymptote
A vertical line on the graph of a rational function where the function approaches infinity or negative infinity, occurring at values that make the denominator zero but not the numerator.
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Horizontal asymptote
A horizontal line that the graph of a rational function approaches as x goes to positive or negative infinity, determined by comparing the degrees of the numerator and denominator polynomials.
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Horizontal asymptote when degrees differ
If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y=0; if equal, it is y equal to the ratio of leading coefficients.
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Hole in the graph
A point where a rational function is undefined due to a common factor in the numerator and denominator that cancels, creating a removable discontinuity at that x-value.
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Removable discontinuity
A hole in the graph of a rational function caused by canceling a common factor, meaning the function could be redefined at that point to make it continuous.
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Rational inequality
An inequality involving one or more rational expressions, solved by finding critical points and testing intervals on a number line to determine where the inequality holds.
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Solving a rational inequality
Find values that make the expression zero or undefined, create a sign chart or test points in the intervals between those values, and determine where the inequality is true.
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Sign analysis for inequalities
A method for solving rational inequalities by determining the sign of the expression in each interval created by critical points, using a chart to track positive and negative regions.
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Test points in inequalities
Values chosen from each interval between critical points to plug into the rational inequality and check whether they satisfy the inequality's conditions.
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Critical points in rational inequalities
The values of x that make the numerator or denominator of a rational inequality zero, which divide the number line into intervals for testing.
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Work-rate problems
Word problems involving rates of work, often expressed as rational expressions, where combined rates are added as fractions to find total time.
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Combined work rates
The sum of individual work rates in a problem, such as two people working together, represented as rational expressions to solve for time.
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Distance problems with rational expressions
Problems involving distance, rate, and time, where equations with rational expressions are used, like average speed over different segments.
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Average speed as rational expression
Calculated as total distance divided by total time, often resulting in a rational expression when distances or times vary.
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Mixture problems with concentrations
Problems where substances are mixed at different concentrations, leading to equations with rational expressions to find amounts or ratios.
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Inverse proportion
A relationship where one quantity is a rational expression inversely proportional to another, such as time and work rate in certain problems.
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Direct proportion with rational functions
When two quantities vary directly, but involve rational expressions, like in scaled mixtures or rates.
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Rational function
A function that can be expressed as a ratio of two polynomials, with graphs that may include asymptotes and holes based on the degrees and factors.
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End behavior of rational functions
The behavior of a rational function as x approaches positive or negative infinity, often approaching a horizontal asymptote based on the degrees of the polynomials.
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Intercepts of rational functions
The points where the graph crosses the axes, with x-intercepts at roots of the numerator and y-intercept at the function's value when x=0.
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X-intercept of a rational function
The x-value where the graph crosses the x-axis, found by setting the numerator equal to zero and solving, provided it doesn't make the denominator zero.
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Y-intercept of a rational function
The y-value where the graph crosses the y-axis, found by substituting x=0 into the function, as long as the denominator is not zero at that point.
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Asymptotic behavior
How a rational function approaches its asymptotes, either vertically or horizontally, which helps in sketching the graph.
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Long division for rational expressions
A method used when the degree of the numerator is greater than or equal to the denominator, to divide polynomials and express the function as a polynomial plus a proper fraction.
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Strategy: Factor first when simplifying
Always begin by factoring the numerator and denominator of a rational expression to identify common factors for cancellation and to avoid errors.
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Common trap: Dividing by zero
A frequent mistake where values that make the denominator zero are not excluded, leading to undefined expressions or incorrect solutions.
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Common trap: Forgetting domain restrictions
Overlooking values that make the denominator zero when solving equations or simplifying, which can result in extraneous solutions.
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Formula: Product of rational expressions
The result of multiplying two rational expressions is a new rational expression formed by multiplying numerators and denominators, then simplifying.
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Formula: Quotient of rational expressions
The result of dividing two rational expressions is obtained by multiplying the first by the reciprocal of the second, followed by simplification.
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Example: Simplify (x^2 - 1)/(x - 1)
Factor the numerator as (x-1)(x+1) and cancel the common factor (x-1), resulting in x+1, but note x ≠ 1 to avoid division by zero.
Original: (x^2 - 1)/(x - 1); Simplified: x + 1 for x ≠ 1
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Example: Add 1/x + 1/y
Find the common denominator xy, rewrite as (y + x)/(xy), and simplify to (x + y)/(xy), excluding x=0 and y=0.
Result: (x + y)/(xy)
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Example: Solve 1/x + 1/y = 1/2
Multiply through by 2xy to get 2y + 2x = xy, rearrange to xy - 2x - 2y = 0, factor, and solve, checking for extraneous solutions like x=0 or y=0.
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Example: Subtract 1/(x+1) - 1/(x-1)
Common denominator (x+1)(x-1), rewrite as [(x-1) - (x+1)] / [(x+1)(x-1)] = (-2) / (x^2 - 1), excluding x=1 and x=-1.
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Example: Multiply (x+1)/x x/(x-1)
Multiply numerators and denominators to get (x+1)x / [x(x-1)], cancel x (if x ≠ 0), resulting in (x+1)/(x-1).
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Example: Divide (x+1)/2 by 1/x
Multiply by the reciprocal: [(x+1)/2] [x/1] = [x(x+1)] / 2, simplifying as needed and excluding x=0.
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Example: Simplify the complex fraction 1/(1/x)
Multiply numerator and denominator by x: 1/(1/x) = x, noting x ≠ 0.
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Example: Solve the inequality 1/x > 2
Subtract 2: 1/x - 2 > 0, common denominator x: (1 - 2x)/x > 0, find critical points x=1/2 and x=0, and test intervals.
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Example: Find vertical asymptote of 1/(x-2)
The denominator is zero at x=2, so there is a vertical asymptote at x=2.
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Example: Work-rate problem with two workers
If A works at 1/3 tank per hour and B at 1/4 tank per hour, together they work at 1/3 + 1/4 = 7/12 tank per hour.
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Example: Distance problem with average speed
For 60 miles at 30 mph and 60 miles at 60 mph, total distance 120 miles, total time 2 + 1 = 3 hours, so average speed is 120/3 = 40 mph.
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Strategy: Check for asymptotes when graphing
Identify vertical and horizontal asymptotes by finding where the denominator is zero and comparing polynomial degrees to understand the graph's behavior.
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Strategy: Use LCD in equations
Multiply both sides of a rational equation by the least common denominator to eliminate fractions, but always verify solutions in the original equation.