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SAI International School

                                            CLASS – VIII, SUB: MATH
                                                LESSON NOTES-9

                                       CHAPTER-14: FACTORISATION
           SUBTOPIC:
                      Division of a polynomial by another polynomial
              1. Division of a polynomial by another polynomial



                  There are many different methods for dividing polynomials, but one thing remains the same: every
                  division will have a divisor (the denominator of the fraction) and the dividend (the numerator of the
                  fraction). Also, in many cases, the same division problem can be solved in multiple ways. To highlight

                  this fact, we're going to do the same problem three different ways, explaining each technique in
                  general and using a example.


           Method 1: Factorization

           In many cases, factoring the numerator and/or the denominator will help simplify matters. Even if this
           technique doesn't completely simplify the fraction, it will often at least lower the degree of the polynomials
           in the numerator and denominator so that we are left with an equivalent expression that's easier to work
           with. It's worth noting that factorization doesn't always work; in particular it will fail if the divisor does not
           evenly divide the dividend (say that three times fast!). But it is usually less-time consuming than the other
           methods when it does work, so it's usually the first thing we try.


           First notice that the numerator and the denominator have a common factor of x.


                                                             頠

                                                         頠    頠ܗ
                                                         頠
                                                             ܗ
           Now we can CANCEL the factors of x, and we are left with


                                                         頠
                                                            頠ʹ   頠ܗ

                                                           ʹ    ܗ
                                                        頠
                                                           頠ʹ   頠ܗ

                                                          ʹ    ܗ




           We are left with a quadratic in the numerator and a linear binomial in the denominator. It turns out we
           can factor the numerator again, as follows:


               頠
                  ܗ  頠  頠ܗ                                                        FACTORISATION USING

                  ʹ  ܗ                                                            SPLITTING OF MIDDLE TERM
                                                    頠
                                                       ܗ      頠    頠ܗ                   SPLIT THE MIDDLE

                                                          ʹ    ܗ                         TERM
                                                                                        USE DISTRIBUTIVE
                                                                                         PROPERTY AND TAKE
                                                                                         THE COMMON GROUP
                                                                                         AS COMMON FACTOR
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