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Page 1 BINOMIAL EXPRESSION An algebraic expression containing only two terms is called a binomial expression. Such as ?? + ?? , 2?? + 5?? , 2?? + 1 ?? , ?? + ?? ?? etc. all are binomial expressions. Page 2 BINOMIAL EXPRESSION An algebraic expression containing only two terms is called a binomial expression. Such as ?? + ?? , 2?? + 5?? , 2?? + 1 ?? , ?? + ?? ?? etc. all are binomial expressions. BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX If ?? , ?? ? ?? , then ??? ? ?? (?? + ?? ) ?? = ?? ?? 0 ?? ?? + ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? + ?? ?? ?? ?? ?? -?? ?? ?? + ? + ?? ?? ?? ?? ?? Here all ?? ?? ?? are called binomial coefficient ?? ?? ?? = ?? ! ?? ! (?? - ?? )! ?? ?? 0 = 1 = ?? ?? ?? ?? ?? 1 = ?? ?? ?? ?? = ?? ?? ?? -?? ; 0 = ?? = ?? Page 3 BINOMIAL EXPRESSION An algebraic expression containing only two terms is called a binomial expression. Such as ?? + ?? , 2?? + 5?? , 2?? + 1 ?? , ?? + ?? ?? etc. all are binomial expressions. BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX If ?? , ?? ? ?? , then ??? ? ?? (?? + ?? ) ?? = ?? ?? 0 ?? ?? + ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? + ?? ?? ?? ?? ?? -?? ?? ?? + ? + ?? ?? ?? ?? ?? Here all ?? ?? ?? are called binomial coefficient ?? ?? ?? = ?? ! ?? ! (?? - ?? )! ?? ?? 0 = 1 = ?? ?? ?? ?? ?? 1 = ?? ?? ?? ?? = ?? ?? ?? -?? ; 0 = ?? = ?? Note: (i) ?? is the exponent or index of binomial (ii) Total number of terms is ?? + 1 (iii) The (?? + 1) ?? h term = ?? ?? +1 = ?? ?? ?? ?? ?? -?? ?? ?? (iv) Sum of exponents of ?? and ?? in any term = ?? (v) Powers of ?? decreasing by 1 and powers of ?? increasing by 1 in subsequent terms (vi) In any term suffix of binomial coefficient is power of ?? (a) Replacing ?? with -?? in the expansion of (?? + ?? ) ?? , we get (?? - ?? ) ?? = ?? ?? 0 ?? ?? - ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 - ? + (-1) ?? ?? ?? ?? ?? ?? (b) Replacing ?? by 1 and ?? with ?? , we get (1 + ?? ) ?? = ?? ?? 0 + ?? ?? 1 ?? + ?? ?? 2 ?? 2 + ? … + ?? ?? ?? ?? ?? (c) (1 - ?? ) ?? = ?? ?? 0 - ?? ?? 1 ?? = ?? ?? 2 ?? 2 - ? … + (-1) ?? ?? ?? ?? ?? ?? (d) (?? + ?? ) ?? + (?? - ?? ) ?? = 2[ ?? ?? 0 ?? ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? … ] = 2[ Sum of terms at odd places ] last term ?? ?? ?? ?? ?? if ?? is even and ?? ?? ?? -1 ?? ?? ?? -1 if ?? is odd (e) (?? + ?? ) ?? - (?? - ?? ) ?? = 2 [Sum of terms at even places] last term ?? ?? ?? -1 ?? ?? ?? -1 if ?? is even and ?? ?? ?? ?? ?? if ?? is odd Page 4 BINOMIAL EXPRESSION An algebraic expression containing only two terms is called a binomial expression. Such as ?? + ?? , 2?? + 5?? , 2?? + 1 ?? , ?? + ?? ?? etc. all are binomial expressions. BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX If ?? , ?? ? ?? , then ??? ? ?? (?? + ?? ) ?? = ?? ?? 0 ?? ?? + ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? + ?? ?? ?? ?? ?? -?? ?? ?? + ? + ?? ?? ?? ?? ?? Here all ?? ?? ?? are called binomial coefficient ?? ?? ?? = ?? ! ?? ! (?? - ?? )! ?? ?? 0 = 1 = ?? ?? ?? ?? ?? 1 = ?? ?? ?? ?? = ?? ?? ?? -?? ; 0 = ?? = ?? Note: (i) ?? is the exponent or index of binomial (ii) Total number of terms is ?? + 1 (iii) The (?? + 1) ?? h term = ?? ?? +1 = ?? ?? ?? ?? ?? -?? ?? ?? (iv) Sum of exponents of ?? and ?? in any term = ?? (v) Powers of ?? decreasing by 1 and powers of ?? increasing by 1 in subsequent terms (vi) In any term suffix of binomial coefficient is power of ?? (a) Replacing ?? with -?? in the expansion of (?? + ?? ) ?? , we get (?? - ?? ) ?? = ?? ?? 0 ?? ?? - ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 - ? + (-1) ?? ?? ?? ?? ?? ?? (b) Replacing ?? by 1 and ?? with ?? , we get (1 + ?? ) ?? = ?? ?? 0 + ?? ?? 1 ?? + ?? ?? 2 ?? 2 + ? … + ?? ?? ?? ?? ?? (c) (1 - ?? ) ?? = ?? ?? 0 - ?? ?? 1 ?? = ?? ?? 2 ?? 2 - ? … + (-1) ?? ?? ?? ?? ?? ?? (d) (?? + ?? ) ?? + (?? - ?? ) ?? = 2[ ?? ?? 0 ?? ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? … ] = 2[ Sum of terms at odd places ] last term ?? ?? ?? ?? ?? if ?? is even and ?? ?? ?? -1 ?? ?? ?? -1 if ?? is odd (e) (?? + ?? ) ?? - (?? - ?? ) ?? = 2 [Sum of terms at even places] last term ?? ?? ?? -1 ?? ?? ?? -1 if ?? is even and ?? ?? ?? ?? ?? if ?? is odd General term in the expansion of (?? + ?? ) ?? The (?? + 1) th term = ?? ?? +1 is called the general term. ?? ?? +1 = ?? ?? ?? ?? ?? -?? ?? ?? Page 5 BINOMIAL EXPRESSION An algebraic expression containing only two terms is called a binomial expression. Such as ?? + ?? , 2?? + 5?? , 2?? + 1 ?? , ?? + ?? ?? etc. all are binomial expressions. BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX If ?? , ?? ? ?? , then ??? ? ?? (?? + ?? ) ?? = ?? ?? 0 ?? ?? + ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? + ?? ?? ?? ?? ?? -?? ?? ?? + ? + ?? ?? ?? ?? ?? Here all ?? ?? ?? are called binomial coefficient ?? ?? ?? = ?? ! ?? ! (?? - ?? )! ?? ?? 0 = 1 = ?? ?? ?? ?? ?? 1 = ?? ?? ?? ?? = ?? ?? ?? -?? ; 0 = ?? = ?? Note: (i) ?? is the exponent or index of binomial (ii) Total number of terms is ?? + 1 (iii) The (?? + 1) ?? h term = ?? ?? +1 = ?? ?? ?? ?? ?? -?? ?? ?? (iv) Sum of exponents of ?? and ?? in any term = ?? (v) Powers of ?? decreasing by 1 and powers of ?? increasing by 1 in subsequent terms (vi) In any term suffix of binomial coefficient is power of ?? (a) Replacing ?? with -?? in the expansion of (?? + ?? ) ?? , we get (?? - ?? ) ?? = ?? ?? 0 ?? ?? - ?? ?? 1 ?? ?? -1 ?? + ?? ?? 2 ?? ?? -2 ?? 2 - ? + (-1) ?? ?? ?? ?? ?? ?? (b) Replacing ?? by 1 and ?? with ?? , we get (1 + ?? ) ?? = ?? ?? 0 + ?? ?? 1 ?? + ?? ?? 2 ?? 2 + ? … + ?? ?? ?? ?? ?? (c) (1 - ?? ) ?? = ?? ?? 0 - ?? ?? 1 ?? = ?? ?? 2 ?? 2 - ? … + (-1) ?? ?? ?? ?? ?? ?? (d) (?? + ?? ) ?? + (?? - ?? ) ?? = 2[ ?? ?? 0 ?? ?? + ?? ?? 2 ?? ?? -2 ?? 2 + ? … ] = 2[ Sum of terms at odd places ] last term ?? ?? ?? ?? ?? if ?? is even and ?? ?? ?? -1 ?? ?? ?? -1 if ?? is odd (e) (?? + ?? ) ?? - (?? - ?? ) ?? = 2 [Sum of terms at even places] last term ?? ?? ?? -1 ?? ?? ?? -1 if ?? is even and ?? ?? ?? ?? ?? if ?? is odd General term in the expansion of (?? + ?? ) ?? The (?? + 1) th term = ?? ?? +1 is called the general term. ?? ?? +1 = ?? ?? ?? ?? ?? -?? ?? ?? Middle term in the expansion of (?? + ?? ) ?? Case 1. ?? = ?? ?? ; ?? ? ?? Then, number of terms in the expansion = ?? + 1 = 2?? + 1; Thus middle term is ?? ?? +1 i.e. ( ?? 2 + 1) ?? h term ?? ?? +1 = ?? ?? ?? 2 ?? ?? 2 · ?? ?? 2 Case 2. ?? = ?? ?? + ?? ; ?? ? ?? Then, number of terms in the expansion = ?? + 1 = 2?? + 2. There are two middle terms, viz. ?? ?? +1 and ?? ?? +2 . i.e. ( ?? +1 2 ) ?? h and ( ?? +3 2 ) ?? h term ?? ?? +1 = ?? ?? ?? -1 2 · ?? ?? +1 2 · ?? ?? -1 2 ?? ?? +2 = ?? ?? ?? +1 2 · ?? ?? -1 2 · ?? ?? +1 2Read More
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