Find the term independent of x in the expansion of (x^4-2/3x)^100
To find the term independent of x, we need to focus on the constant term in the expansion. The constant term is obtained by choosing the x terms which cancel each other out when multiplied together.
We can represent the constant term as:
C(100, k) * (x^4)^k * (-2/3x)^(100-k)
where k is the number of times we choose the x^4 term out of the 100 brackets.
For the constant term, we need to have k times the x^4 term and (100-k) times the -2/3x term.
Therefore, we need to solve the following equation:
4k - (2/3)(100-k) = 0
simplifying, we get:
k = 25
So the constant term is:
C(100,25) * (x^4)^25 * (-2/3x)^(75)
= C(100,25) * x^100 * (-2/3)^75
Since we are only interested in the constant term, we can ignore the x^100 term and we are left with:
C(100,25) * (-2/3)^75
which is a numerical value and is the term independent of x.
To find the term independent of x in the expansion of (x^4 - 2/3x)^100, we need to look for the term that does not contain any x.
The general form of the binomial expansion is given by:
(x + y)^n = C(n,0) * x^n * y^0 + C(n,1) * x^(n-1) * y^1 + C(n,2) * x^(n-2) * y^2 + ... + C(n,n) * x^0 * y^n
In this case, x^4 - 2/3x = (x^4) * (1 - (2/3)x)^100, so y = 1 and x = (2/3)x.
Using the binomial theorem, the term independent of x will occur when the exponent of x is 0. In other words, we want to find the term with (2/3x)^0.
The term with (2/3x)^0 will have the form:
C(100,k) * (x^4)^100-k * ((-2/3x)^0)^k
Since anything raised to the power of 0 is 1, ((-2/3x)^0)^k will always be 1. Therefore, the term independent of x will be:
C(100,k) * (x^4)^100-k * 1^k
For this term to be independent of x, the exponent of x in (x^4)^100-k must be 0. This means that 100 - k = 0, which implies k = 100.
Substituting k = 100 into the expression for the term, we have:
C(100,100) * (x^4)^0 * 1^100
= C(100,100) * 1 * 1^100
= C(100,100)
The term independent of x in the expansion of (x^4 - 2/3x)^100 is therefore given by the coefficient C(100,100) or "100 choose 100". The value of this coefficient is 1.
So, the term independent of x is equal to 1.