4. Rational Algebraic Expressions
Note You need to understand how to multiply algebraic expressions using the distributive law before starting work on this tutorial. If you feel you need to review this, go back to 3. Multiplying and Factoring Algebraic Expressions.
Q What is a Rational Expression?
Rational Expression
A rational expression is an algebraic expression of the form P/Q, where P and Q are simpler expressions (usually polynomials), and the denominator Q is not zero.
A rational number is any number that can be written in the form a/b, where a and b are integers and b ≠ 0. it is necessary to exclude 0 because the fraction represents a ÷ b, and division by zero is undefined.
A rational expression is an expression that can be written in the form P/Q where P and Q are polynomials and the value of Q is not zero. Some examples of rational expressions:
5/3; (x^2 + 1)/2; 7/(y 1); (ab)/c; [(a^2)(b]/c^2; (z^2 + 3z + 2)/ (z + 1) ect. Like a rational number, a rational expression represents a division, and so the denominator cannot be 0. A rational expression is undefined for any value of the variable that makes the denominator equal to 0. So we say that the domain for a rational expression is all real numbers except those that make the denominator equal to 0. Examples:
1) x/2
Since the denominator is 2, which is a constant, the expression is defined for all real number values of x.
2) 2/x
Since the denominator x is a variable, the expression is undefined when x = 0
3) 2/(x  1)
x  1 ≠ 0
x ≠ 1
The domain is {x x ≠ 1}. Or you can say:
The expression is undefined when x = 1.
4) 2/(x^2 + 1)
Since the denominator never will equal to 0, the domain is all real number values of x.
Algebra of Rational Expression
Rule Example
Multiplication:
P
Q  R
S = PR
QS
 
x + 1
x  (x  1)
2x + 1 = (x  1)(x + 1)
x(2x + 1) = x2  1
x(2x + 1)

Addition with Common Denominator:
P
Q + R
Q = P + R
Q  
 
y
xy + 1 + x  1
xy + 1 = x + y  1
xy + 1

General Addition Rule:
(works with or without common denominator)
P
Q + R
S = PS + RQ
QS
 
y
x + x  1
y = y2 + x(x  1)
xy

Subtraction with Common Denominator:
P
Q  R
Q = P  R
Q  
 
y
x2  1  x  1
x2  1 = x + y +1
x2  1

General Subtraction Rule:
(works with or without common denominator)
P
Q  R
S = PS  RQ
QS
 
y2
x  xy
y + 1 = y2(y + 1)  x2y
x(y + 1)

Reciprocals:
1
 P
Q 

 =
 Q
P


    
 
1
 x + 1
y  1 

 =
 y  1
x + 1


    

Cancellation:
PR
QR = P
Q  
 
y2(xy  1)
x(xy  1) = y2
x  

Multiplication
Multiplying Rational Expressions (page 1 of 2)
With regular fractions, multiplying and dividing is fairly simple, and is much easier than adding and subtracting. The situation is much the same with rational expressions (that is, with polynomial fractions). The only major problem I have seen students having with multiplying and dividing rationals is with illegitimate cancelling, where they try to cancel terms instead of factors, so I'll be making a big deal about that as we go along. Remember how you multiply regular fractions: You multiply across the top and bottom. For instance:
And you need to simplify, whenever possible:
While the above simplification is perfectly valid, it is generally simpler to cancel first and then do the multiplication, since you'll be dealing with smaller numbers that way. In the above example, the 3 in the numerator of the first fraction duplicates a factor of 3 in the denominator of the second...