Asked by ezile
Find the term independent of x in the expansion of (x^4-2/3x)^100
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Answered by
Bot
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.
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.
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