1.4 Fractional Expressions

Fractional Expression- a quotient of two algebraic expressions

Rational Expression- a quotient p/q of two polynomials p and q (think RATIO)

Algebraic Expressions – No = sign

                I.e.: x+5, 2x² -3x+4, and so on, not set to zero

• Simplifying rational expressions:

• A common nonzero factor in the numerator and denominator of a quotient may be canceled. 

• ·         A rational expression is simplified, or reduced to lowest terms, if the numerator and denominator have no common integral factors greater than 1

• ·         We often use properties of quotients to obtain one rational expression

• ·         To add or subtract two rational expressions, we usually find a common denominator and use the following properties of quotients

• ·         We usually use the Least Common Denominator (LCD) when adding or subtracting quotients.  To find the LCD, we proceed as follows

• ·         We can use Prime Factorization, by factoring each denominator into primes and then forming the product of the different prime factors, using the LARGEST exponent that appears with each prime factor

• ·         The prime factorizations of the denominators 24 and 18 are 24=2³ ·3 and 18= 2· 3³

• ·         To find the LCD, we form the product of the different prime factors, using the larges exponent associated with each factor, giving us 2³·3²

• ·         Sums and differences of rational expressions

• ·         The LCD is x²(3x-2).  To obtain three quotients having the denominator x²(3x-2), we multiply the numerator and denominator of the first quotient by x, those of the second by x², and those of the third by 3x-2, which gives us

• ·         Complex Fraction- a quotient in which the numerator and/or the denominator is a fractional expression

• ·         To solve, we take the numerator of the given expression into a single quotient and then use a property for simplifying quotients

• ·         An alternative method is to multiply the numerator and denominator of the given expression by (x+3)(a+3), the LCD of the numerator and denominator, and then simplify the result

• ·         Some quotients that are not rational expressions contain denominators of the form a+√b or √a+√b

• ·         These quotients can be simplified by multiplying the numerator and denominator by the conjugate a-√b or √a-√b, respectively

• ·         Of course, if a-√b appears, multiply by a+√b instead

• ·         Sometimes it is necessary to rationalize the numerator

•  In calculus, it is of interest to determine what is true if h is very close to zero, in which case we obtain the following

• ·         However this is a meaningless expression.  If we use the rationalized form, however, we obtain the following information

• ·         Simplifying a fractional expression