**Unformatted text preview: **Binomial Theorem An algebraic expression containing two terms is called binomial expression. Example
( x+a)
(1/2 +x)
(2/x -1/x3) The general form of the binomial expression is (x+a) and expansion (x+a)n n? N is called Binomial
expression It was developed by Sir Issac newton The general expression for the Binomial Theorem is (x+a)": "C0 x"a0 + "C1 x"‘1a1 + "C2 x"‘za2 + .................. + "C, x""ar ....... + "Cn x°a" (x + a)’I = Z;o(:)x""‘a" Proof:
We can prove this theorem with the help of mathematical induction Let us assume P(n) be the statement is (x+a)": "Co x"a° + "C1 x""a1 + "CZ x"'2a2 + .................. + "cr x""ar ....... + nCn x‘Jan Step 1 Now the value of P(1) (ma)1 = ‘00 x‘ao +101 x“‘a1
=(x+a) So P(1) is true Step 2 Now the value of P(m) (x+a)"‘ = "‘00 x'"a° + "‘01 xm‘a‘ + "‘C2 x'"‘2a2 + .................. + "‘C,x'“"a' ....... + "‘C" xoa'“
Now we have to prove
(x+a)""1 = ””00 x“'”a‘J + ""101 Xma‘ + ””02 x""1a2 ++ "‘*‘C,x"“"‘ar ....... + ""‘Cmﬂ x"a""+1 Now (x+a)"”1 =(x+a)(x+a)'“ = (x+a)("‘co x'"a° + l“C1)("""a‘ + "‘02 X'Ma2 + .................. + ""C,x“""ar ....+ "'0" xoa'“)
="1c0 x"“‘1aO +( m01+mco)x'“a‘ + ("'02 +'“C1)x""‘az + ......... +( ”Cm + m0...) x‘a'“ + "‘0", x"a"‘+1 As "‘c,.1 + mcr = ""10, SO ='"*‘Cox"‘"‘a° +""‘C1x'“a1 +"""‘sz"“‘a2 + .................. ~*"‘*‘C,x"""“ar ....... + ""10",” xoam'ﬂ ...

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- Spring '13
- Rahul CHawla
- Math, Binomial Theorem, Binomial