381. If in equation (1) of the preceding article we suppose b=c=d=a, we obtain (x+a)4=24+4ax]+6a_x2 + 4aRx+at. We shall now employ the same method to prove a formula known as the Binomial Theorem, by which any binomial of the form xta can be raised to any assigned positive integral power. 382. To find the expansion of (x+a)" when n is a positive integer. Consider the expression (x+a) (a+b) (x+c)......(2+k), the number of factors being n. The expansion of this expression is the continued product of the n factors, x+a, x+b, x + c.......X+k, and every term in the expansion is of n dimensions, being a product formed by multiplying together n letters, one taken from each of these n factors. The highest power of x is un, and is formed by taking the letter x from each of the n factors. The terms involving 2n-1 are formed by taking the letter x from any n-1 of the factors, and one of the letters a, b, c,...k from the remaining factor; thus the coefficient of xn-1 in the final product is the sum of the letters a, b, C.......k; denote it by Si The terms involving 2n–2 are formed by taking the letter x from any n- 2 of the factors, and two of the letters a, b, c,...k from the two remaining factors; thus the coefficient of 2n-2 in the final product is the sum of the products of the letters a, b, C...k taken two at a time; denote it by Sz. And, generally, the terms involving x-r are formed by taking the letter x from any n- r of the factors, and r of the letters a, b, c,...k from the r remaining factors; thus the coefficient of anar in the final product is the sum of the products of the letters a, b, c,...k taken i at a time; denote it by Sp The last term in the product is abc...k; denote it by Sin = x+8,21-1+S.2.2"-2+...+$.&n- + ... +87-1X+SIn S, the number of terms is n; in S, the number of terms is the same as the number of combinations of n things 2 at a time; that is, “C2; in Sg the number of terms is nCz; and so on. Now suppose b, c,...k, each equal to a; then S, becomes nCja; S, becomes "Cya?; Sg becomes nCzQ3; and so on; thus (x+a)" =2+Ca.2-1+Cja*x*–2+"Czasun-3 + ... +nCran; + substituting for nC1, "C2)...we obtain 1.2 apan-3+...tan, 1.2.3 the series containing n+1 terms. This is the Binomial Thcorem, and the expression on the right is said to be the expansion of (x+a)". 383. The coefficients in the expansion of (x+a)are very conveniently expressed by the symbols "C, "C2, C3,..."Cn. We shall, however, sometimes further abbreviate them by omitting n, and writing C1, C2, C3,...On With this notation we have (x+a)"=+Caxn-1+Cyazn-2+Cga3.21-3+...+Cra. If we write – a in the place of a, we obtain (2-a)= 2+C (-a).xn-1+C,(-a)2rn-2 +Cg (-a)34-3+...+on(-a)" = 39 - Carn-1+ C a 2n-2-Cga3.40-3+...+(-1)"Cyan. Thus the terms in the expansion of (x+a)" and (x – a)" are numerically the same, but in (a -a)n they are alternately positive, and negative, and the last term is positive or negative according as n is even or odd. Example 1. Find the expansion of (oc+y). =x6 + 6xby + 15x4y2 + 20.xoy3 + 15xy4 +6xyó + yo, on calculating the values of 6C7, 6C2, C3, .......... Example 2. Find the expansion of (a – 2x)”. (a – 2x)?=a? – ?C (2x) + ?C9a5 (2.c)2 – 7Ca4(2x)3 + to 8 terms. Now remembering that "Cr=nCn-n, after calculating the coefficients up to 7C3, the rest may be written down at once; for 7C,=7C3; 7C5=7C,; and so on. Hence 7.6 7.6.5 1.2 1.2.3 - 21a? (2.c)5 + 7a (2x)6 – (2x)? =a? – 14abac+84am - 280a4ac3 + 560a3x4 - 672a2m +448ax6 – 128x7. 384. In the expansion of (x+a)”, the coefficient of the second term is nCı; of the third term is "C2; of the fourth term is "C'z ; and so on; the suffix in each term being one less than the number of the term to which it applies; hence nC, is the coefficient of the (r+1)th term. This is called the general term, because by giving to y different numerical values any of the coefficients may be found from "Cr; and by giving to x and a their appropriate indices any assigned term may be obtained. Thus the (r+1) term may be written n (n-1) (n-2)...(n-1+1) nCylin-ra, or c="a". La In applying this formula to any particular case, it should be observed that the index of a is the same as the suffix of C, and that the sum of the indices of x and a is n. Example 1. Find the fifth term of (a +2.23)17. a13 (22%) x 16a1313 1.2.3.4 = 38080a13x72. Example 2. Find the fourteenth term of (3-a) 15. The required term = 15C135)-(-a)18 = 15C,x(-9a15) [Art. 370.] =-945a13. 385. The simplest form of the binomial theorem is the expansion of (1+x)". This is obtained from the general formula of Art. 382, by writing 1 in the place of x, and x in the place of a. Thus (1+x)"=1+"Cx+Cox2 + +nCpX* + +nCnxn n(n-1) 1.2 r 386. The expansion of a binomial may always be made to depend upon the case in which the first term is unity; thus y )" xzwhere z=? = {«(1+%) }"=>"(1+2)", wł * Called “factorial p" and meaning the continued product of all the integers from 1 to r. Thus4=1.2.3.4= 24. Example. Find the coefficient of x16 in the expansion of (r? – 2x) 10. We have (x2 – 2x)10=x*' (1 (1-2)"; 2\10 and, since x20 multiplies every term in the expansion of 1 we have in this expansion to seek the coefficient of the term which 1 contains x4 Hence the required coefficient=10C4 (-2) 10.9.8.7 1.2.3.4 =3360. x 16 fat1 PROOF BY MATHEMATICAL INDUCTION. 387. By actual multiplication we obtain the following identities : (a+x)=a2 + 2ax+x?, (a+x)4=a4 +-4aRx + 6a2x2 +4ax3 + x4. Selecting any one of these and rewriting so as to exhibit the laws of formation of exponents and coefficients, we have 4 4.3.2 a4-22+ -a4-873 1.2.3 aox+(ao=1 Art. 238). 1.2.3.4 Assume that these laws of formation hold for (a +x)", n being any positive integer. Then (a +x)" =an + nan-lx+ n(n-1) an–2x 1.2 ..(1). 1.2.3 Multiplying each side of the identity by (a+x) and combining terms, we obtain, (a+x)*+1=a*+1+(n+1)a"x+ 1.2 ..... (2). 1.2.3 It will be seen that n in (1) is, in every instance, replaced by (n+1) in (2). Hence if the theorem be true for any value of n, it will be true for the next higher value. We have shown + n(n+1) an-122 an—243 by multiplication that the theorem is true when n successively n(n-1) 1.2 We observe that in any term (1) The exponent of x, the second term of the binomial, is one less than the number of the term from the first. (2) The sum of the exponents is n. (3) The last factor of the denominator is the same as the exponent of the second term of the binomial. (4) The numerator contains as many factors as the denominator, the first being n, and each succeeding one decreasing Hence the (r+1)th or general term of (a +x)" is n(n-1)... for r factors an-X?. 1.2.3.. Example. Find the 6th term in the expansion of (2a+6)10. Here n=10, and r+1=6. 10.9.8.7.6 3.2.7.6 We have (2a)565 (2a)566 1.2.3.4.5 1 = 252(2)5a5b5 = 80640555. by 1. ...r EXAMPLES XXXIX. a. Expand the following binomials: 1. (x+2)4 2. (x+3) 5. 4. (a—«)5. 5. (1-2y)5. Write down and simplify: 10. The 4th term of (1+x) 12. 11. The 6th term of (2-y). 12. The 5th term of (a–56). 13. The 15th term of (2x— 1)17. 14. The 7th term of (1 15. The 6th term of ( 3x+ 16. Find the value of (x-V3)++ (x + V3)4. 9 (1-)" |