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High School Mathematics - 2
7.3 Binomial Theorem

Coefficient of a particular power of x
  1. Let the particular power occur in the (r+1)th term.
  2. Write the (r+1)th term of the given binomial.
  3. Equate the power of x in the (r+1)th term and the given power.
  4. Evaluate r and substitute in in step 2.
Eample: Find the co-efficient of x152)10.
Solution:
  1. Let x15 occur in the (r+1)th term.
  2. Tr+1 = 10Crx10-r .(-x2)r.
  3. 10Cr.x10-r.x2r.(-1)r.
  4. = (-1)r.10Crx10+r
  5. x10+r = x15
  6. r = 5
Terms Independent of x
Procedure
  1. Let (r+1)th term be the term independent of x.
  2. 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:

  1. Let (r+1)th term be independent of x.
  2. Tr+1 = 9Cr.(3/2x2)9-r.(-1/3x)r
  3. Putting 18-3r = 0, we get r = 6.
  4. 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:

  1. Let Tr+1 have the greatest coefficient.
  2. Then coefficient of Tr+1 >= the coefficient of T.
  3. 10Cr >= 10Cr-1
  4. (10-r+1)/r >= 1 or 11 >= 2r, or r <= 51/2
  5. Hence upto r = 5 the coefficients increase, the greatest of them being r = 5.
  6. The greatest coefficient = 10C5 = 252
Answer : 252


Directions: Answer the following
Q 1: Find the coefficient of a6b3 in the expansion of (2a-b/3)9
Answer:

Q 2: Find the term containing x2 if any in (3x-1/2x)8.
Answer:

Q 3: Find numerically the greatest term in the expansion of (5x+4)7 when x = 1(write 1st, 2nd etc)
Answer:

Q 4: Find the greatest term in the expansion of (2-3x)7 when x = 1.
Answer:

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Question 6: This question is available to subscribers only!


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