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### High School Mathematics - 27.3 Binomial Theorem

 Coefficient of a particular power of x Let the particular power occur in the (r+1)th term. Write the (r+1)th term of the given binomial. Equate the power of x in the (r+1)th term and the given power. Evaluate r and substitute in in step 2. Eample: Find the co-efficient of x152)10. Solution: Let x15 occur in the (r+1)th term. Tr+1 = 10Crx10-r .(-x2)r. 10Cr.x10-r.x2r.(-1)r. = (-1)r.10Crx10+r x10+r = x15 r = 5 Terms Independent of x Procedure Let (r+1)th term be the term independent of x. Put the power of x in this term equal to zero and evaluate it. Example: Find the term independent of x in the expansion of (3/2x2 - 1/3x)9. Solution: Let (r+1)th term be independent of x. Tr+1 = 9Cr.(3/2x2)9-r.(-1/3x)r Putting 18-3r = 0, we get r = 6. The required term is (-1)6.9C6. 3-3/23 = 9C3.(1/33.23 = 7/18 Answer: 7/18 Example: Find the greatest coefficient in the expansion of (1+x)10 Solution: Let Tr+1 have the greatest coefficient. Then coefficient of Tr+1 >= the coefficient of T. 10Cr >= 10Cr-1 (10-r+1)/r >= 1 or 11 >= 2r, or r <= 51/2 Hence upto r = 5 the coefficients increase, the greatest of them being r = 5. The greatest coefficient = 10C5 = 252 Answer : 252 Directions: Answer the following
 Q 1: In the expansion (x2+1/x)n, the coefficient of the 4th term is equal to the coefficient of the ninth term Find n.Answer: Q 2: Find numerically the greatest term in the expansion of (2+3x)7 when x = 4/5Answer: Question 3: This question is available to subscribers only! Question 4: This question is available to subscribers only!