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How do I find the first three terms, in descending powers of X, in the expansion of (x- 2/X⁾⁶

User Lawinslow
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Final answer:

To find the first three terms of the expansion (x - 2/x)^6, apply the binomial theorem: x^6 for the first term, -12x^3 for the second term, and -60x for the third term. These are computed using the binomial coefficients and the powers of x and -2/x.

Step-by-step explanation:

To find the first three terms in descending powers of X in the expansion of (x - 2/x)^6, we need to use the binomial theorem, which expresses the expansion of a binomial raised to a positive integer power. The binomial theorem states that (a+b)^n = a^n + n*a^(n-1)*b + n(n-1)/2*a^(n-2)*b^2 + ..., where 'n' is the power and 'n choose k' represents the binomial coefficient for the k-th term in the expansion.

In our case, the binomial (x - 2/x) raised to the power of 6 means we have 'a' as x and 'b' as -2/x. So, the first term (a^n) is x^6. The second term involves the binomial coefficient of 6 choose 1, which is 6, so 6*x^(6-1)*(-2/x). Finally, the third term involves the binomial coefficient of 6 choose 2, which is 15, so 15*x^(6-2)*(-2/x)^2.

The first three terms are then calculated as:

  1. x^6
  2. -6*x^4*(2/x) = -12*x^3
  3. +15*x^2*(-2/x)^2 = -60*x

So, the first three terms of the expansion in descending powers of X are x^6, -12x^3, and -60x.

User Emilis Vadopalas
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