Topic 8.1 (Rational Expressions I)

To determine the values of x that make the rational expression undefined, factor  the denominator and set each factor equal to zero and solve.

Sample.         (x + 5)(x - 3) = (x + 5)(x - 3) Factor the denominator and set each

                           x² - x - 12       (x - 4)(x + 3)           factor equal to  0.

                             x - 4 = 0    and    x + 3 = 0

                                  x = 4                       x = -3          {-4, 3}     Ans.  

_______________________________________________________________________________

One of the fundamental theorems of Algebra is that all fractions must be reduced to it lowest terms.

Always reduce completely unless otherwise instructed.

All fractions must be completely factored, as only factors can be reduced.

Sample:       x² + 5x + 6 Factor both numerator and denominator completely first.

                       x² + 6x + 9

                      (x + 2)(x + 3)  Cross out the common factors in the numerator and the

                      (x + 3)(x + 3)     denominator.

                              x + 2        Ans.

                              x + 3

________________________________________________________________________________

Sample:        2x² - 32         Factor both numerator and denominator completely first.

                   4x² - 44x + 112

                   2(x + 4)(x - 4)    Cross out the common factors in the numerator and the

                   4(x - 4)(x - 7)         denominator.

                            x + 4             Ans.

                          2(x - 7)

_______________________________________________________________________________

Sample:       Reduce to lowest terms:     3a²b⁵    Reduce the 3 and the 27.

                                                                          27ab⁷     Cross out one a in the numerator

                                                                                             and one a in the denominator.

                                                                                           Cross out 5 b’s in the numerator

                                                                                             and 5 b’s in the denominator.

                                                                                    a      Ans.

                                                                                   9b²

^{___________________________________________________________________________________________}

To divide fractions, invert the divisor (the fraction on the right of the division sign) and multiply.

To multiply fractions, cross out common (like terms) in the numerator and the denominator, commonly called cancellation or reducing through divison.

_______________________________________________________________________________

Sample problem:           15a²b² · 2cd³    Reduce the 2’s, the 15 and the 5.

                                             2c³d      5ab²    Reduce the a’s, the b’s, the c’s, and the d’s

                                                     3ad²               Ans.

                                                        c²

________________________________________________________________________________

Sample problem:     3ab²  x  6a²b     Invert the divisor (the fraction on the right of

                                      13d⁴      11d²        the division sign) and multiply.

                                     3ab²   x  11d² Reduce to lowest terms

                                     13d⁴       6a²b

                                               11b                       Ans.

                                              26ad²

To add or subtract fractions, change all denominators to the Lowest Common Denominator (LCD) and add or subtract as indicated.

_____________________________________________________________________________

Sample problem:     3x  +  18x          The denominators are the same, so just add the

                                      5y       5y                 numerators and reduce if necessary.

                                          21x                             Ans 

                                           5y

________________________________________________________________________________

Sample problem:    4y + 6    -    3y + 4       The denominators are the same, so just

                                     3y + 6         3y + 6              subtract the numerators.

                                  (4y + 6) - (3y + 4)

                                             3y + 6

                                       4y + 6 - 3y - 4            Remember the negative sign changes the

                                             3y = 6                         sign of both terms in the parenthesis to

                                                                                 the right of it.

                                               y + 2    =        1            Ans.

                                            3(y + 2)             3