Algebra 2 - Simplifying Rational Expressions


  • Just like rational numbers (which are expressed in the form of pq, where p and q are integers and  q0), rational expressions are simply ratios of two polynomials.
  • Mathematically, a rational expression in variable 'x' can be represented as f(x)g(x), in which the value of g(x) can never be zero.
  • Some examples of rational expressions include: 2x2+4x+17x-541-x2x3-5x+3, etc.

Simplifying Rational Expressions

  • In general, simplifying any rational expression can be accomplished in two steps.
    1. Factorize the polynomials of the numerator and denominator
    2. Reduce the expression by canceling out common factors

Solved Examples

Example 1: Is 1+x2x-5 a rational expression?

Answer: No, because the numerator is not a polynomial.


Example 2: Is 2x-9 as rational expression?

Solution: Yes, because 2x-9 can be expressed as 2x-91 which satisfied the definition of a rational expression.


Example 3: Simplify x2-16x+4.

Solution: x2-16x+4=x+4·x-4x+4=x-4


Example 4: Simplify 2xx2+2x.

Solution: 2xx2+2x=2xx·x+2=2x+2


Example 5: Simplify 3x-1-3x2-2x+1.

Solution: 3x-1-3x2-2x+1=3x-1-3x2+2x-1=3x-1-3x-1·x+1=1-x+1=-1x+1


Example 6: Simplify x2+4x-5x2-5x+4.

Solution: x2+4x-5x2-5x+4=x-1x+5x-1x-4=x+5x-4

Cheat Sheet

  • To simplify any rational expression, factorize the polynomials in the numerator and denominator and cancel out the common factors.

Blunder Areas

  • A rational expression 2xx-3 is not defined for x=3.