A rational expression is a fraction whose numerator and denominator are polynomials, such as x + 3x − 2 or 5x2. The denominator cannot equal 0, so values that make it 0 are excluded. You simplify, add, subtract, multiply and divide rational expressions the same way you work with numerical fractions.
In Spanish
expresión racional
Also called
algebraic fraction, polynomial fraction, fraction with variables
Factor first, then cancel the common factor x − 2. The simplified form equals the original everywhere except at the excluded values x = 2 and x = −3.
Formal definition and notation
A rational expression is an expression of the form p(x)q(x), where p and q are polynomials and q is not the zero polynomial. It is defined for every value of the variable except those that make q(x) = 0. Rational expressions form a system like the rational numbers: sums, differences, products and quotients by nonzero rational expressions are again rational expressions.
A rational number is a ratio of integers, like 27. A rational expression is a ratio of polynomials, like 2x + 7; it becomes a rational number when you substitute a value.
Rational Expressionvs.Rational function
A rational expression is a fraction to simplify or combine. A rational function, f(x) = p(x)q(x), uses that fraction as an input-output rule and has a graph.
Rational Expressionvs.Rational exponent
A rational exponent is a fraction in the exponent, like x12. A rational expression has variables in a fraction, like 1x2.
Quick check: 3 questions
Question 1 of 3
What are the excluded values of x + 1x2 − 16?
Show answer and explanation
Answer: D.x2 − 16 = (x − 4)(x + 4) is 0 at 4 and −4. x = −1 makes the numerator 0, which is allowed; 16 forgets the square root; “4 only” misses the negative root.
Question 2 of 3
Simplify x + 5x2 + 5x.
Show answer and explanation
Answer: A. Factor the bottom: x(x + 5). Cancel x + 5 to get 1x, with x ≠ 0, −5. The other choices cancel terms instead of factors.
Question 3 of 3
What is x3 · 6x2?
Show answer and explanation
Answer: C. Multiply: 6x3x2, then cancel 3 and x: 2x. The first choice loses the x on top, 2x flips the power, and the last adds instead of multiplying.
Rational Expression: frequently asked questions
How do you simplify a rational expression?
Factor the numerator and the denominator completely, then cancel any factor that appears in both. For x2 − 252x + 10, factor to (x − 5)(x + 5)2(x + 5) and cancel x + 5 to get x − 52. State the excluded values from the original denominator: here x ≠ −5.
Why are some values excluded from a rational expression?
Because division by zero is undefined. Any value of the variable that makes the denominator 0 gives an expression with no meaning, so it is excluded. For 4x − 7, substituting 7 would mean 40. The excluded values stay excluded even after you simplify the expression.
How do you add rational expressions with different denominators?
Find the least common denominator, rewrite each fraction with it, then add the numerators. For 1x − 1 + 2x, the LCD is x(x − 1), so the sum is x + 2(x − 1)x(x − 1) = 3x − 2x(x − 1). This is the same method you use for 14 + 16.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-APR.D.6, CCSS HSA-APR.D.7
Cite this entry
Effortless Math Team. “Rational Expression.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/rational-expression/
Flashcard
TermRational Expressionp(x)q(x), q(x) ≠ 0
DefinitionA fraction of two polynomials. x2 − 9x + 3 simplifies to x − 3 for x ≠ −3.
FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Rational NumberA rational number is any number that can be written as a fraction of two integers, ab, where b is not 0. Fractions, integers, terminating decimals, and repeating decimals are rational numbers: 34, −5 = −51, 1.6 and 0.333… all qualify. Numbers such as √2 and π are irrational.Number System · Grade 7, Grade 8, Algebra 1DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3QuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5Least Common DenominatorThe least common denominator (LCD) of two or more fractions is the smallest number that every denominator divides into evenly. It is the least common multiple of the denominators. For 14 and 56, the LCD is 12, so you rewrite them as 312 and 1012 before you add, subtract or compare.Number System · Grade 5, Grade 6, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2Common DenominatorA common denominator is a number that can be used as the denominator of two or more fractions at once. It is any common multiple of the denominators. For 13 and 14, 12 works: 412 and 312. You need one to add, subtract or compare fractions with different denominators.Number System · Grade 4, Grade 5, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Rational FunctionA rational function is a function that can be written as a fraction of two polynomials, f(x) = p(x)q(x), where q(x) is not the zero polynomial. It is undefined wherever q(x) = 0, so its graph often has vertical asymptotes or holes. Examples: f(x) = 1x and g(x) = x + 2x − 5.Functions · Algebra 2, PrecalculusAsymptoteAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1x has both: x = 0 and y = 0.Functions · Algebra 2, Precalculus, CalculusPowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x2 + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 68 = 6 ÷ 28 ÷ 2 = 34. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5, Grade 6