How to simplify rational expressions?
Type one rational expression, or several joined by + − * and /. The page combines them into one fraction, factors the top and the bottom, cancels the common factors, and lists the values the variable cannot take.
- Simplified
- (x - 3)/(x + 1)
(x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).
- Excluded values
- x ≠ -3, x ≠ -1
- Numerator, factored
- (x - 3) (x + 3)
- Denominator, factored
- (x + 1) (x + 3)
Simplified: (x - 3)/(x + 1). (x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).
How to calculate
Simplifies, adds, subtracts, multiplies and divides rational expressions in one variable: one fraction in lowest terms, with the excluded values.
Example with the default inputs (Rational expression (x^2 - 9)/(x^2 + 4x + 3)): (x^2 - 9)/(x^2 + 4x + 3) simplifies to (x - 3)/(x + 1).
Method: Combine into one fraction (multiply across, or over the product of the denominators), factor the top and the bottom, and cancel the common factors.
- One variable; whole-number powers.
- The simplified form equals the expression wherever the expression is defined.
Worked examples
Each example is checked against the calculator on every build.
- Rational expression (x^2 - 9)/(x^2 + 4x + 3) gives Simplified (x - 3)/(x + 1), Excluded values x ≠ -3, x ≠ -1.Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 1: (x − 3)/(x + 1)
- Rational expression (x^2 + 4x - 5)/(3x + 18) * (2x - 1)/(x + 5) gives Simplified (x - 1) (2x - 1)/(3 (x + 6)), Excluded values x ≠ -6, x ≠ -5.Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 2: (x − 1)(2x − 1)/(3(x + 6))
- Rational expression (2x^2 + x - 6)/(x^2 - 1) / ((x^2 - 4)/(x^2 + 2x + 1)) gives Simplified (2x - 3) (x + 1)/((x - 1) (x - 2)), Excluded values x ≠ -2, x ≠ -1, x ≠ 1, x ≠ 2.Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 3: (2x − 3)(x + 1)/((x − 1)(x − 2))
- Rational expression 6/(x^2 + 4x + 4) - 2/(x^2 - 4) gives Simplified 4 (x - 4)/((x - 2) (x + 2)^2), Excluded values x ≠ -2, x ≠ 2.Source: OpenStax, College Algebra 2e, section 1.6 Rational Expressions (https://openstax.org/books/college-algebra-2e/pages/1-6-rational-expressions), Example 5: 4(x − 4)/((x + 2)²(x − 2))
How it works
The expression uses one letter, numbers, + − * /, brackets and whole-number powers. Anything else (sqrt, sin, a fractional power) has no answer.
- One fraction. The expression is written as p/q: a product of fractions multiplies tops and bottoms; a quotient multiplies by the reciprocal; a whole-number power (A/B)^n is A^n/B^n; a sum or difference A/B ± C/D becomes (AD ± CB)/(BD).
- Factor. A computer algebra system (nerdamer, open source) expands and factors each part of the product p, and of q, on its own; a power of a part is factored once and raised to that power. The top and bottom shown are these factored products, before cancelling. The algebra’s answers are checked numerically. A factored form with a number that is not a whole number or a whole-number fraction has no answer.
- Cancel. Each factor that appears in both, written the same way, is cancelled to the lower of its two powers. The two whole-number factors are divided by their greatest common divisor, and a minus sign moves to the top.
- Check. The result must equal the typed expression to 10⁻⁹ at 15 fixed test points (such as −7.31, −0.91, 0.21, 1.33, 8.93) where both are defined.
- Excluded values come from every denominator in the typed expression, nested ones too (1/(1/(x − 3)) excludes 3): the real solutions of each distinct factor of the top of each denominator = 0 from the algebra (factored as in step 2), each checked by substitution, in increasing order; "none" when no denominator holds the letter or none has a real zero. If the algebra cannot solve one of them, the list is not shown.
Sums inside a factor are written highest power first, and factors of a product lowest degree first.
Worked examples by hand
(x² − 9)/(x² + 4x + 3) (OpenStax College Algebra 2e, Example 1). = (x − 3)(x + 3)/((x + 1)(x + 3)) = (x − 3)/(x + 1), x ≠ −3, −1.
(x² + 4x − 5)/(3x + 18) × (2x − 1)/(x + 5) (Example 2). = (x + 5)(x − 1)(2x − 1)/(3(x + 6)(x + 5)) = (x − 1)(2x − 1)/(3(x + 6)).
(2x² + x − 6)/(x² − 1) ÷ (x² − 4)/(x² + 2x + 1) (Example 3). = (2x − 3)(x + 2)(x + 1)²/((x − 1)(x + 1)(x − 2)(x + 2)) = (2x − 3)(x + 1)/((x − 1)(x − 2)).
6/(x² + 4x + 4) − 2/(x² − 4) (Example 5). Over (x² + 4x + 4)(x² − 4): the top is 6(x² − 4) − 2(x² + 4x + 4) = 4x² − 8x − 32 = 4(x − 4)(x + 2), the bottom (x + 2)³(x − 2). Cancelling one x + 2 gives 4(x − 4)/((x − 2)(x + 2)²).
Other questions people ask
What is a rational expression?
A fraction whose top and bottom are polynomials, such as (x² − 9)/(x² + 4x + 3). It is undefined where the bottom is 0.
How do I simplify a rational expression?
Factor the numerator and the denominator, then cancel the factors they share. (x² − 9)/(x² + 4x + 3) = (x − 3)(x + 3)/((x + 1)(x + 3)) = (x − 3)/(x + 1), for x ≠ −3 and x ≠ −1.
How do I multiply or divide rational expressions?
To multiply, multiply the tops and the bottoms, then simplify. To divide, multiply by the reciprocal of the second expression. Type them with * and /, and put the second fraction in brackets when dividing: (2x^2 + x - 6)/(x^2 - 1) / ((x^2 - 4)/(x^2 + 2x + 1)).
How do I add or subtract rational expressions?
Write each over a common denominator, then add or subtract the numerators. The page multiplies the denominators together and then cancels, so the result is the same as with the least common denominator: 6/(x² + 4x + 4) − 2/(x² − 4) = 4(x − 4)/((x − 2)(x + 2)²).
What are the excluded values?
The values of the variable that make a denominator of the original expression 0. They stay excluded after simplifying, even when the factor that caused them has cancelled: (x − 3)/(x + 1) above still needs x ≠ −3.
Can I use more than one variable?
No. The page works in one letter, such as x or t. For 5/x + 6/y, combine by hand: (5y + 6x)/(xy).
How is the answer checked?
The simplified form must equal the typed expression at 15 test points where both are defined. Each excluded value is a solution from the algebra, checked by putting it back into the denominator.