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ABOUT ME O BTech in Computer Science u MBA in Finance (IBS Hyderabad) 10 Year Corporate Experience. More than 80 Courses, 1.5K+ Lessons & 32K + Followers Passion to learn and teach online. 6+ Years of Teaching Experience for IT Maths Completed 7-certifications from IVY League Universities

BINOMIAL THEOREM BINOMIAL EXPRESSION Any algebraic expression which contains two dissimilar terms is called binomial expression For example x y, xy1, BINOMIAL THEOREM The formula by which any positive integral power of a binomial expression can be expanded in the form of a series is known as BINOMIAL THEOREM. + 3 etc. 1/3 x z x -y) If x, y e R and ne N, then : ( x + y Cox" + C1Xn-1 y + C2xmp + + C,x"y'+, + Cy= crx" ryr r-0

Few Binomial expansions Consider two numbers a and b, then a+b a2 +2ab +b2 a+b (a+b)(a+b) (a+b).(a+b) (a+bf-( + 2ab + b2y@+2ab + b2) -(a+b) a +2ab+b2 a3 + 3a2b + 3ab2 + b3 - (a2 +2ab +b a2 +2ab +b -a4 +4a3b+6a2b2 +4ab3 +b.

Some Examples Illustration: Expand (y + 2). -y12y60y4 160y3 240y2 + 192y 64. Illustration: Expand the following binomial 4 3x3 Sol: By using formula of binomial expansion.

2 4 3x 3x 3x 0 27 2 6 27 2 81 ,12 16 DEDUCTION FROM BINOMIAL THEOREM Results of Binomial Theorem D-1 On replacing a by -a, in the expansion of (xa), we get r-0 Therefore, the terms in (x-a)n are alternatively positive and negative, and the sign of the last term is positive or negative depending on whether n is even or odd.

D-2 Putting x = 1 and a = x in the expansion of (x + a. we get r-0 This is the expansion of (1 + x)" in ascending powers of x. D-3 Putting a = 1 in the expansion of (x + a)", we get r-0 This is the expansion of (1x) in descending powers of x. D-4 Putting x = 1 and a =-x in the expansion of (x + ay, we get

D-5 From the above expansions, we can also deduce the following Illustration: Write first 4 terms of 1 2 2y Sol:

Illustration:If in the expansion of (1x)m (1 x), the coefficients of x and x2 are 3 and 6 respectively then m is (A) 6 (B) 9 (C) 12 (D) 24 (1 + x)" (1 -x)" = | 1 + mx + (m)(m -1).x 2 Sol : 2 Coefficient of x = m-n = 3 n(n 1) m(m1) Coefficietof x2mn 2 2 Solving) and (), we get m=12and n = 9.