LEMMA

Rational expressions

Rational-expression operations

ALEKS placement

Multiply and divide by factoring and canceling; add and subtract by finding an LCD.

What this covers

  • Factor every numerator and denominator. Record where any original denominator is zero.
  • For division, the entire second fraction must also be nonzero. Then multiply by its reciprocal: turn only that second fraction upside down.
  • For addition, build the LCD from the highest power of each distinct factor.
  • Combine the numerators, then simplify and state restrictions.

Worked example

Worked example

Add 3/x + 2/(x + 1)

  1. x=0x=−1​The original denominators cannot be zero. These restrictions stay with the result.
  2. LCD=x(x+1)The least common denominator contains each needed factor. Both fractions can be rewritten with this denominator.
  3. x(x+1)3(x+1)​+x(x+1)2x​Multiply the top and bottom of each fraction by its missing factor. Multiplying by one preserves each value.
  4. x(x+1)3(x+1)+2x​The denominators now match. Add the numerators and keep that common denominator.
  5. x(x+1)5x+3​Distribute and combine the like terms on top. The original excluded inputs remain excluded.
Another worked example

Subtract 3/x − 2/(x + 1)

  1. x=0x=−1​Both original denominators must be nonzero. Keep these restrictions throughout.
  2. x(x+1)3(x+1)​−x(x+1)2x​Multiply each numerator and denominator by its missing factor to make the denominators match.
  3. x(x+1)3(x+1)−2x​Subtract the entire second numerator and keep the common denominator.
  4. x(x+1)x+3​Distribute and combine the like terms on top. Both original restrictions remain.

A common mistake

A mistake Lemma catches

From 3/x+2/(x+1)

3/x+2/(x+1)=2x+15​3/x+2/(x+1)=x(x+1)5x+3​

“Tops and bottoms were added separately: (3 + 2)/(x + (x + 1)) = 5/(2x + 1). Fractions add only over a common denominator: over x(x + 1) the tops are 3(x + 1) and 2x, which add to 5x + 3. Check at x = 1: 3/1 + 2/2 = 4, but 5/3 is not 4.”

Try one

Sample problem

Combine into one fraction.

x1​+y1​

Practice rational operations free

No account, no email. Every line you type is checked by the same computer algebra system as the worked example above.

Where this fits

Before this

After this