Binomial Theorem Cheat Sheet 
                              
                             Section                                                                                                                                               *Remarks 
                             Introduction  1.  Binomial is about EXPANSION of expression.                                                                                          First of all, start with knowing the 
                                                                                                                                                                                   definitions and formulae 
                                                                       n       n ! n $ n−1 1 ! n $ n−2 2 ! n $ n−3 3
                                                       2.  (a+b) =a +#                     &a b +#               &a b +#                &a b +... 
                                                                                   # 1 &                 # 2 &                   # 3 &
                                                                                   "       %             "       %               "      %
                                                                                                ! n $ n−r r
                                                       3.  General Term: T =#                           &a b  
                                                                                         r+1    # r &
                                                                                                "       %
                                                                                                    ! n $             n!
                                                       4.  Binomial Coefficient: #                          &=                  
                                                                                                    # r &        r!(n−r)!
                                                                                                    "       %
                             Easy                      1. Using Binomial Theorem, find the first four terms of  1+3x 4.                                                            Easy Questions usually deal with 
                             Questions                                                                                                                (         )                  applying formulae in a very 
                                                                                                                                                                                   straightforward 
                             Standard                                                                                                                               7              Standard Questions are questions 
                             Questions                                                                                                                    "      x%                often found in our textbook 
                                                       1.  Using Binomial Theorem, find the first three terms of $                                                 ' .  
                                                                                                                                                            1−
                                                                                                                                                          $     5'
                                                                                                                                                          #        &               exercises or assessment books. 
                                                            i)  Hence, obtain the coefficient of x2 in the expansion                                                               Usually, the values of the 
                                                                                           7                                                                                       variables are changed but the 
                                                                                 "     x%                                                                                          question types will not divert too 
                                                                 of              $        ' .  
                                                                     (3x+2) 1−
                                                                                 $     5'                                                                                          much away from these standard 
                                                                                 #        &
                                                            ii)  Hence, estimate the value of                                             7, correct to 3                          form.   
                                                                                                                        2.3×(0.98)
                                                                 significant figures. 
                                                                                                                                                                         7
                                                                                                                                                          !         2 $
                                                       2.  Using Binomial Theorem, find the first three terms of #5x+                                                   & . 
                                                                                                                                                          #         x2 &
                                                                                                                                                          "             %
                                                            Hence, obtain the coefficient of x2 in the expansion 
                                                                                             7
                                                                            2 "         2 %
                                                            of(4x−2) $5x+                  ' . 
                                                                              $         x2 '
                                                                              #            &
                                                                                                                           !         3$7
                                                       3.  Find the term independent of x in #x+ x3 &  
                                                                                                                           "           %
                                                       4.  Find the coefficient of x−1 in the binomial expansion of 
                                                                             7
                                                             "     2     2 %
                                                             $3x −          '  
                                                             $          x3 '
                                                             #              &
                                                       5.  Given that the coefficient of the third term in the expansion of 
                                                                         nis                , find the value of n where n is a positive 
                                                             (2x−3)           -253440
                                                            integer. 
                                                       6.  In the expansion of                              n, the coefficient of x2and x3are in 
                                                                                                (3+2x)
                                                            the ratio of 9:1. Find the value of n. 
                                                                   
                             Challenging  1.  Given that  1+kx n =1+20x+45k2x2+..., find the value of k and                                                                        Challenging questions are hardly 
                             Questions                                           (         )                                                                                       predictable. They are either 
                                                            n.                                                                                                                     more complicated in form or 
                                                       2.  Obtain the first three terms in the expansion, in ascending                                                             require knowledge from other 
                                                                                               x2                                                                                  chapters. To answer these 
                                                                                                    6                                                           x3
                                                            powers of x, of (4− 3 ) . Hence, find the coefficient of                                               in the          questions, students have to be 
                                                                                           2x           2x                                                                         generally strong in their 
                                                                                                 6            6                                                                    foundation in math and have a 
                                                            expansion of (1+                    ) (1−        ) . 
                                                                                             3             3                                                                       thorough understanding of the 
                                                                                                                                                                                   chapter.