Find the terms as stated: (x-1/x)^6 constant term
there will be no constant term.
The (x^3)(1/x)^3 term will be constant.
That is the 4th term of the expansion.
So, what is C(6,4)?
Oops. The 4th term is C(6,3)
To find the constant term in the expansion of the expression (x-1/x)^6, we need to expand it using the binomial theorem and identify the term with zero exponent of x.
1. First, let's expand the expression (x-1/x)^6 using the binomial theorem. According to the binomial theorem, the expansion of (x+y)^n is given by:
(x+y)^n = nC0 * x^n * y^0 + nC1 * x^(n-1) * y^1 + nC2 * x^(n-2) * y^2 + ... + nCn * x^0 * y^n,
where nCk represents the binomial coefficient given by n! / (k!(n-k)!).
In our case, x is the variable, and y is -1/x. Thus, our expression becomes:
(x-1/x)^6 = 6C0 * x^6 * (-1/x)^0 + 6C1 * x^5 * (-1/x)^1 + 6C2 * x^4 * (-1/x)^2 + ... + 6C6 * x^0 * (-1/x)^6.
2. Simplifying the expression:
- The term 6C0 * x^6 * (-1/x)^0 is equal to 6C0 * x^6 * (-1)^0 = 6 * x^6 * 1 = 6x^6.
- The term 6C1 * x^5 * (-1/x)^1 is equal to 6C1 * x^5 * (-1/x) = 6 * x^5 * (-1) * (1/x) = -6x^4.
- The term 6C2 * x^4 * (-1/x)^2 is equal to 6C2 * x^4 * (-1/x)^2 = 6 * x^4 * (1/x^2) = 6/x^2.
3. Continuing this pattern, we see that the exponent of x decreases by one, and the exponent of (1/x) increases by one in each subsequent term. Therefore, the term with a constant coefficient does not contain any x or (1/x) terms.
Hence, the constant term in the expansion (x-1/x)^6 is obtained when the exponent of x is 0 and the exponent of (1/x) is 6. This means the constant term is 6/x^6.