Binomial Theorem. 137 



13. Theorem. The sum of all the coefficients in the expansion 

 of (x-\-a)" equals 2". 



In the expansion of (x-\-a)"^\xt x=i and a=\. We then 



have 



n(n—i) n(n—i)(?i—2) 



2 1-2.3 



14. Examples. 



7. Expand (x-\-af. 

 2. Expand (d—c)\ 



J. Expand O'-f 3/- 



^. Expand (b'—(^)^. 



5. Expand (x-\-af. 



6. Expand (x-{-2c)^. 



7. Expand (2>b+^)\ 



8. Expand (x'-\-a^)\ 

 g. Expand (2ax—x^f. 



10. Expand W ab—^abf- 



^2-\ 6 



11. Expand y+— I . 



I J J 



12. Expand (5—ix)'\ 



ij. Find the 5th term of ( xj' -\- x"- )" . 



f -?- 3. 



14.. Find the 9th term of [flr^ + f-t" 



/f. Find the 7^th term oi \n"-\ I . 



\ n"\ 



16. Expand ( x^ -\- 2ax -\- a" )^ . 



ly. Expand (a/<^^— 2 Jf)'^ 



18. Find the 1000 term in (x-^a)^'""'. 



15. Expansion OF A Polynomial. The power of a polynomial 

 can be obtained in the following manner. Suppose it is required 

 to expand (a-\-b-^c)^. We can proceed thus : 



(a + b+c)'=[a + (b+c)J 

 = a' + 2>^^(b + c) + 2>^(b+c)^-\r(b-\-c)\ 

 which, when the powers of <^ + <r are developed, becomes 



a^-f- 3^^'<^-f 3aV4- 3«<^'+ 6^7/^rH- 3^^" + ^-1- 3^V+ 3<5r" -hf ^ 

 Notice that the result is a homogeneous syuwietrical functio7i of 

 a, b, and c. 



—17 



