Coefficient of a Particular Term

Coefficient of a Term in the Binomial Expansion

  • The binomial theorem provides a systematic way to expand expressions of the form (a+b)n, where 'n' is a non-negative integer and 'a' and 'b' are any real or complex numbers.

  • The expansion consists of multiple terms, each of which includes a coefficient and powers of 'a' and 'b.'

  • The coefficient of a particular term represents the constant factor that multiplies that specific term in the expanded expression.

Finding the Coefficient of a Specific Term

To find the coefficient of a particular term in the binomial expansion, follow these steps:

  1. Identify the Term: Determine which term in the binomial expansion you want to find the coefficient for. Specify the term using its exponents of 'a' and 'b.'

  2. Use the Binomial Theorem: Apply the binomial theorem formula to find the coefficient. The general formula is:

    (a+b)n = C(n, 0) * an * b0 + C(n, 1) * a(n-1)) * b1 + C(n, 2) * a(n-1) * b2 + ... + C(n, n) * a0 * bn

    • 'n' is the non-negative integer.
    • 'C(n, k)' represents the binomial coefficient, which is used to find the coefficient of a specific term.
    • 'a' and 'b' are the constants.
  3. Identify the Coefficient: After applying the binomial theorem formula, you can find the coefficient of the term you're interested in.

Example 1:

Given the binomial expansion (x+y)5, find the coefficient of the term with 'x2 * y3.'

  • The term you're interested in is 'x2 * y3.'

  • Apply the binomial theorem formula:

    (x+y)5 = C(5, 0) * x5 * y0 + C(5, 1) * x4 * y1 + C(5, 2) * x3 * y2 + C(5, 3) * x2 * y3 + C(5, 4) * x1 * y4 + C(5, 5) * x0 * y5

  • The coefficient of the term 'x2 * y3' is C(5, 3) = 10.

Example 2:

In the binomial expansion (2a-3b)4, find the coefficient of the term with 'a2 * b2.'

  • The term you're interested in is 'a2 * b2.'

  • Apply the binomial theorem formula:

    (2a-3b)4 = C(4, 0) *(2a)4 * (-3b)0 + C(4, 1) * (2a)3 * (-3b)1 + C(4, 2) * (2a)2 * (-3b)2 + C(4, 3) * (2a)1 * (-3b)3 + C(4, 4) * (2a)0 * (-3b)4

  • The coefficient of the term 'a2 * b2' is C(4, 2) = 6.