Rational Expressions

A rational expression is an algebraic fraction whose numerator and denominator are polynomials. (Remember: a polynomial can be a monomial or a sum of monomials.)

Remember: a rational number is a number determined by the ratio of an integer,( -4, -3, -1, 0, 2, 3, 5), and a natural non-zero number, ( 1,2,3,4,5, etc.)

7/3 is a rational number.

y/x³  is a rational expression.

Excluded values of the rational expression are any values of variables that result in the denominator being zero.

For (5a + 2) / x² - 6x + 8, the excluded values are x = 4, and x = 2.

Simplify Rational Expressions:

Eliminate any common factors in the numerator and denominator.

(6n + 36) / (n²  - 36) =

6(n + 6) / (n +6) (n - 6) =

6/ (n - 6)

Multiply Rational Expressions:

Multiply the numerators and the denominators. Eliminate the common factors.

(x + 4)  (x + 8)   · (x + 1) (x - 7)   =

(x - 8) (x + 1)       (x + 2) (x + 4)

 

(x + 8) (x - 7)

(x - 8) (x +2)

 

Dividing Rational Expressions:

 

To divide, multiply the first expression by the inverse of the second expression.

 

2a + 2  ÷  4a+ 4  =

 a + 3         a + 5

 

2a + 2  ·    a + 5  =

 a + 3        4a + 4

 

2(a + 1) · a + 5         =

 a + 3       4(a + 1)

 

2(a + 5)   =

4(a + 3)

 

(a + 5)

2(a + 3)

 

Divide a Polynomial by a Monomial:

 

Divide each term in the polynomial by the monomial, and then simplify.

 

 x² + 20x + 24   =     x² + 20x + 24   =  x  + 5  + 6

         4x                    4x     4x      4x        4            x

 

 

Divide a Polynomial by a Polynomial:

 

Use long division:

 

( 3x² + 9x² + x - 5) ÷ (x + 3) =

 

             3x²           + 1                          

 x + 3 ) 3x³ + 9x² + x - 5

             3x³  +9x²              

                                 x -  5

                                 x + 3

                                     - 2                    

 

  = 3x² + 1 -     2           

                      x + 3 

 

Add/Subtract Rational Expressions with Like Denominators:

Add or subtract the numerators and simplify the result.

   2x       +    8        =     2x + 8   =  2 (x + 4)   = 2

 x + 4         x + 4             x + 4          x + 4

 

 

 m - 16  -  (- 3m + 6)  =     m - 16 + 3m -6   =   4m - 22   = 2(m + 11)  = m +11  

2m + 4      2m + 4             2m + 4                      2m + 4        2( m+ 2)        m + 2

 

 

Add/Subtract Rational Expressions with Unlike Denominators:

Find the Least Common Denominator, reform the fractions, then add or subtract the numerators, and then simplify the result.

   - x - 3              +        x + 6       =

  x² + 6x + 9                x + 3

 

   - x - 3              +        x + 6       =

  (x + 3)(x + 3)           x + 3

 

   - x - 3              +   ( x + 6) (x + 3)       =

  (x + 3)(x + 3)          ( x + 3) (x + 3)

 

 - x - 3 + x² + 9x +18       = 

  (x + 3)(x + 3)            

 

  x² + 8x +15       =    (x + 5) (x + 3)    =  (x + 5)

  (x + 3)(x + 3)           ((x + 3) (x + 3)       (x + 3) 

 

Mixed Expressions:

An expression that contains the sum of a monomial and a rational expression.

x +       5

        (x + 4)

 

 

Complex Fraction:

 

A fraction that has one or more fractions in the numerator or the denominator.

 

   x

   y

 ____

  m

   n

 

To solve, multiply the numerator by the reciprocal of the denominator:

 

  x   · n     = xn

  y     m       ym