Binominal Notes

BINOMIAL BINOMIAL 1. BINOMIAL EXPRESSIONS An algebraic expression containing two terms is called a binomial expression. For example , 2x + 3, x2–x/3, x + a etc. are Binomial Expressions. 2. BINOMIAL THEOREM The rule by which any power of a binomial can be expanded is called the Binomial Theorem. 3. BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX If x and a are two real numbers and n is a positive integer then (x + a)n = nC0 xna0 + nC1xn–1a + nC2xn–2a2 +…..... + nCrxn–rar +......+ nCnx0an. Where nC0, nC1, nC2, nC3,......nCr ..... are called binomial coefficients which can be denoted by C0, C1, C2, C3, .......Cr ...... General Term : In the expansion of (x+a)n, (r+1)th term is called the general term which can be represented by Tr +1. Tr + 1= nCr xn–r ar = nCr( first term)n–r (second term)r. Characteristics of the expansion of (x + a)n Observing to the expansion of (x + a)n, n N, we find that(i) The total number of terms in the expansion = (n + 1) i.e. one more than the index n. (ii) In every successive term of the expansion the power of x (first term) decreases by 1and the This expansion can be obtained by putting (–a) in place of a in the expansion of (x+a)n. General term = (r + 1)th term Tr+1= nCr( –1)r. xn–r ar (ii) By putting x = 1 and a = x in the expansion of (x + a) n , we get the following result (1+x)n = nC0 + nC1 x + nC2 x2+ ....+ nCr xr + .......+ n Cn xn which is the standard form of binomial expansion. General term = (r + 1)th term Tr+1 = nCr x r = (iii) By putting (–x) in place of x in the expansion of (1+ x)n (1–x) n = n C 0– n C 1 x + nC 2 x 2 – nC 3 x 3 + ..... + (–1)r nCrx r +......+ nCnxn. General term = (r + 1)th term Tr+1 = (–1)r. nCr x r of x and a is equal to n (index). (iii) The binomial coefficients of the terms which are at equidistant from the beginning and from the end are always equal i.e. Cr = nCn-r n (n  1)(n  2).......(n  r  1) . xr r! = (–1)r . 4. NUMBER OF TERMS IN THE EXPANSION OF (x + y + z)n (x + y + z)n can be expanded as(x + y + z )n = {(x + y) + z }n = (x + y)n + nC1(x + y)n–1.z + nC2(x + y)n– 2 z2 + ......+ nCn zn. = (n + 1) terms + n terms + (n–1) terms + .........+ 1 term  Total number of terms = (n +1) + n + (n –1) + ....+ 1 power of (second term) increases by 1. Thus in every term of the expansion, the sum of the powers n(n  1)(n  2).......(n  r  1) r .x r! = (n  1)(n  2) 2 5. MIDD LE TERM IN THE EXPANSION OF (x + a)n (a)If n is even, then the number of terms in the expansion i.e. (n+1) is odd, therefore, there will n Thus (iv) C0 = nCn, nC1 = nCn–1, n C2 = nCn–2 etc. n Cr–1 + nC r = n+1Cr n Some deduction of Binomial Theorem : (i) Expansion of (x-a)n. (x-a) n = nC 0 x na 0 – nC 1x n–1 a 1 + n C 2x n-2a 2– n C 3 x n–3 a 3 + .....+ (–1) r n C r x n–r a r + ......+ (–1)n nCn xo an n  2  be only one middle term which is   2  n  i.e.   1 2  th term. th n  so middle term =   1 term. 2   th term. BINOMIAL (b)If n is odd, then the number of terms in the expansion i.e. (n +1) is even, therefore there will be two middle terms which are  n  1  =   2  th th n  3  term. and   2  Note : (i) When there are two middle terms in the expansion then their Binomial coefficients are equal. (ii) Binomial coefficient of middle term is the greatest Binomial coefficient. 6. TO DETERMINE A PARTICULAR TERM IN THE EXPANSION n 1    In the expansion of  x    , if xm occurs in x   Tr+1, then r is given by n – r ( + ) = m  r= n  m  Thus in above expansion if constant term i.e. the term which is independent of x, occurs in Tr+1 then r is determined by n – r ( + ) = 0 n  r=  7. TO FIND A TERM FROM THE END IN THE EXPANSION OF (x+a)n It can be easily seen that in the expansion of (x+a)n. (r+1)th term from end = (n-r+1) th term from beginning. i.e. Tr+1(E) = Tn–r+ 1 (B)  Tr(E) = Tn–r+2 (B) 8. BINOMIAL COEFFICIENTS & THEIR PROPERTIES In the expansion of (1+ x) n ; i.e.(1+ x) n = nC0+ n C1x + ........+ nCr xr + ....+ nCnxn The coefficients nC0, nC1 ,nCn of various powers of x, are called binomial coefficients and they are written as C0, C1, C2,......,Cn Hence (1+x)n = C0+C1x + C2x2+.....+ Crxr + ......+ Cnxn ..(1) Where C0 = 1, C1 = n, C2= Cr = n(n  1) 2! n(n  1)......(n  r  1) , Cn = 1 r! Now, we shall obtain some important expressions involving binomial coefficients(a)Sum of Coefficient : putting x = 1 in (1), we get C0+ C1 + C2 + ......+ Cn = 2n ...(2) (b)Sum of coefficients with alternate signs : putting x = –1 in(1) We get C0- C1 + C2 – C3 + ...... = 0 ...(3) (c) Sum of coefficients of even and odd terms: from (3), we have C0+ C2+ C4 + .......= C1+ C3+ C5 + ........ ...(4) i.e. sum of coefficients of even and odd terms are equal. from (2) and (4)  C0+ C2 + ........= C1+ C3 + .......= 2n–1 9. APPLICATIONS OF BINOMIAL THEOREM (a) With the help of binomial theorem, we can find out the value of sq. root, cube root and 4th root etc. of the given number upto any decimal places. (b) To find the sum of Infinite series : We can compare the given infinite series with the expansion of (1 + x) n = 1 + nx + n(n  1) 2! x2 + ..... and by finding the value of x and n and putting in (1 + x)n the sum of series is determined.