A rational expression is a fraction with polynomials on top and bottom, like (x + 2)/(x² − 4). Every rule you use with number fractions still applies. The extra work is factoring, and keeping track of values that make the denominator zero.
Excluded values
A denominator can never be zero. Before you simplify, find the x values that make the original denominator zero. Those are excluded from the domain, even if the factor cancels later.
Example State the excluded values of (x + 1)/(x² − 5x + 6)
- Factor the bottom: (x − 2)(x − 3).
- It is zero at x = 2 and x = 3.
- Excluded values: x ≠ 2, x ≠ 3.
Simplifying
Factor the top and the bottom completely, then cancel common factors. The factoring guide covers every pattern you’ll need.
Example Simplify (x² − 9)/(x² + 5x + 6)
- Factor: (x − 3)(x + 3) / ((x + 2)(x + 3)).
- Cancel x + 3: (x − 3)/(x + 2).
- Excluded values from the original: x ≠ −2, x ≠ −3.
Canceling terms instead of factors
(x + 6)/(x + 2) does not simplify to 6/2 = 3. You can only cancel things that are multiplied, never pieces that are added. Try x = 0: 6/2 = 3, but at x = 1 you get 7/3.
A factor and its opposite cancel to −1: (x − 5)/(5 − x) = −1, because 5 − x = −(x − 5).
Multiplying and dividing
Multiply: factor everything, cancel any factor on top with any factor on the bottom, then multiply across.
Divide: flip the second fraction and multiply.
Example Simplify (x² − 4)/(3x) · 6x²/(x + 2)
- Factor: (x − 2)(x + 2)/(3x) · 6x²/(x + 2).
- Cancel x + 2, and 6x²/3x = 2x.
- Result: 2x(x − 2), with x ≠ 0 and x ≠ −2.
Example Simplify (x² − x − 6)/(x² − 9) ÷ (x + 2)/(x + 3)
- Flip and multiply: (x² − x − 6)/(x² − 9) · (x + 3)/(x + 2).
- Factor: (x − 3)(x + 2)/((x − 3)(x + 3)) · (x + 3)/(x + 2).
- Everything cancels: 1, with x ≠ 3, −3, −2.
Adding and subtracting
You need a common denominator, just like with number fractions. Use the least common denominator: every factor that appears, at its highest power.
Example Simplify 3/(x + 1) + 2/(x − 1)
- LCD: (x + 1)(x − 1).
- Rewrite: 3(x − 1)/LCD + 2(x + 1)/LCD.
- Add the tops: (3x − 3 + 2x + 2)/LCD = (5x − 1)/((x + 1)(x − 1)).
Example Simplify x/(x² − 4) − 1/(x − 2)
- Factor: x² − 4 = (x − 2)(x + 2). The LCD is (x − 2)(x + 2).
- Rewrite the second fraction: (x + 2)/LCD.
- Subtract: (x − (x + 2))/LCD = −2/((x − 2)(x + 2)).
Not distributing the minus
x − (x + 2) is −2. Writing x − x + 2 = 2 flips the sign of the answer. Put the second numerator in parentheses before you subtract.
Complex fractions
A fraction with fractions inside it simplifies fastest if you multiply the top and bottom by the LCD of all the small fractions.
Example Simplify (1 + 1/x) / (1 − 1/x²)
- The LCD of the small fractions is x². Multiply top and bottom by x².
- Top: x² + x = x(x + 1). Bottom: x² − 1 = (x − 1)(x + 1).
- Cancel x + 1: x/(x − 1).
Solving rational equations
Multiply both sides by the LCD to clear every fraction, solve, then check that no answer is an excluded value.
Example Solve 2/x + 3/(x − 1) = 4/(x(x − 1))
- Excluded: x ≠ 0, x ≠ 1.
- Multiply by x(x − 1): 2(x − 1) + 3x = 4.
- 5x − 2 = 4, so x = 6/5.
- 6/5 isn’t excluded, so x = 6/5.
Example Solve x/(x − 3) = 3/(x − 3) + 2
- Excluded: x ≠ 3.
- Multiply by x − 3: x = 3 + 2(x − 3) = 2x − 3.
- x = 3. But 3 is excluded.
- No solution.
Problems to try
Simplify (x² + 4x)/(x² − 16).
Show answer
x/(x − 4), with x ≠ 4, −4.
Simplify (2x − 6)/(9 − x²).
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−2/(x + 3). The top is 2(x − 3) and the bottom is −(x − 3)(x + 3).
Multiply (x + 3)/(x − 1) · (x² − 1)/(x² + 3x).
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(x + 1)/x.
Simplify 1/x + 1/(x + 2).
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(2x + 2)/(x(x + 2)).
Simplify 5/(x − 2) − 3/(x + 2).
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(2x + 16)/((x − 2)(x + 2)).
Solve 1/x + 1/(2x) = 3.
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x = 1/2. Multiply by 2x: 2 + 1 = 6x.