1. RATIONAL EXPRESSIONS

2. Rational Expressions and Functions: Multiplying and Dividing Objectives –Simplifying Rational Expressions and Functions –Rational Functions –Multiplying –Dividing and Simplifying

3. Definition of a Rational Expression An expression that consists of a polynomial divided by a nonzero polynomial is called a rational expression. Examples of rational numbers:

4. Domain of a Rational Expression The domain of a Rational Expression is the set of all real numbers that when substituted into the Expression produces a real number. If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is: {x | x  2}. illegal if this is zero Note: There is nothing wrong with the top = 0 just means the fraction = 0

5. Finding the Domain of Rational Expression Set the denominator equal to zero. The equation is quadratic. Factor the equation Set each factor equal to zero. Solve The domain is the set of real numbers except 5 and -5 Domain: {a | a is a real number and a ≠ 5, a ≠ -5}

6. Rational Functions Like polynomials, certain rational expressions are used to describe functions. Such functions are called rational functions.

7. The function given by Gives the time, in hours, for two machines, working together, to complete a job that the first machine could do alone in t hours and the second machine could do in t + 5 hours. How long will the two machines, working together, require for the job if the first machine alone would take (a) 1 hour? (b) 6 hours Solution Since division by 0 is undefined, the domain of a rational function must exclude any numbers for which the denominator is 0. The domain of H is

8. Multiplying Products of Rational Expressions To multiply two rational expressions, multiply numerators and multiply denominators:

9. Recall from arithmetic that multiplication by 1 can be used to find equivalent expressions: Multiplying by 2/2, which is 1 3/5 and 6/10 represent the same number Similarly, multiplication by 1 can be used to find equivalent rational expressions: Multiplying by, which is 1, Provided x ≠ -3

10. So long as x is replaced with a number other than -2 or -3, the expressions represent the same number. If x = 4, then and A rational expression is said to be simplified when no factors equal to 1 can be removed.

11. Simplifying Rational Expressions and Functions We “removed” the factor that equals 1 : It is important that a rational function’s domain not be changed as a result of simplifying There is a serious problem with stating that the functions represent the same function. The difficulty arises from the fact that the domain of each function is assumed to be all real numbers for which the denominator is nonzero. Thus, unless we specify otherwise,

12. There is a serious problem with stating that the functions represent the same function. The difficulty arises from the fact that the domain of each function is assumed to be all real numbers for which the denominator is nonzero. Thus, unless we specify otherwise, Domain of F = { x ≠ -2, x ≠ -3}, and Domain of G = { x| x ≠ -2} Thus, as presently written, the domain of G includes -3, but the domain of F does not. This problem is easily addressed by specifying

13. Write the function given by in simplified form. Solution We first factor the numerator and the denominator, looking for the largest factor common to both. Once the greatest common factor is found, we use it to write 1 and simplify: Note that the domain of f = {t |t ≠ 0} Factoring. The greatest common factor is 7t

14. Write the function given by in simplified form, and list all restrictions on the domain. Solution Note that the domain of g = {x |x ≠ -1/2 and 2} Factoring the numerator and the denominator. Rewriting as a product of two rational expressions Remove a factor equal to 1:

15. Let’s try:

16. “Canceling” is a shortcut often used for removing a factor equal to 1 when working with fractions. Canceling removes factors equal to 1 in products. It cannot be done in sums or when adding expressions together. CANCELING Now cancel any like factors on top with any like factors on bottom. Simplify. 1 1 1 1 1

17. Multiply. Then simplify by removing a factor equal to 1 Factoring the numerator and the denominator and finding common factors Multiplying the numerators and also the denominators Remove a factor equal to 1 Simplify

18. Let’s try:

19. Division of Rational Expressions Let p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,

20. To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem. DIVIDING RATIONAL EXPRESSIONS Multiply by reciprocal of bottom fraction.

21. To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem. DIVIDING RATIONAL EXPRESSIONS Multiply by reciprocal of bottom fraction. Factor Notice that (p - 6) and (6 – p) are opposites and form a ratio of -1 Reduce common factors Solution