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• blackholeftw
In summary, the binomial expansion formula is a mathematical formula used to expand binomials into a series of terms. The coefficient of a specific term can be found using the combination formula, and the number of terms in an expansion is equal to the power plus one. The purpose of using binomial expansion is to simplify complex problems and it can be applied by rewriting the expression in the form (a + b)^n and using the formula to expand it into a series of terms.
blackholeftw
As titled, been cracking my head over it.

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Here’s the first part of the question to get you going ...

For $(a-b)^9$, $\displaystyle T_{r+1} = \binom{9}{r+1} a^{9-(r+1)}(-b)^{r+1}$

$T_{r+1} = \dfrac{9!}{(r+1)!(8-r)!} (2x)^{8-r} \left(-\dfrac{1}{2x^2}\right)^{r+1} = \dfrac{9!}{(r+1)!(8-r)!} (-1)^{r+1} \cdot 2^{7-2r}x^{6-3r}$

Multiplying this term by $x^h$ would yield the variable factor as $x^{6-3r+h} = x^{-3} \implies 3r-h=9$

## 1. What is binomial expansion?

Binomial expansion is a mathematical technique used to expand a binomial expression raised to a power. It involves using the binomial theorem to find the coefficients of each term in the expanded expression.

## 2. How do I expand (2x-1/(2x^2))^9?

To expand this expression, you can use the binomial theorem or Pascal's triangle. First, write out the power of 9 as (9 choose 0), (9 choose 1), (9 choose 2), etc. Then, substitute these values into the binomial theorem formula or use Pascal's triangle to find the coefficients. Finally, multiply the coefficients by the terms in the expression to get the expanded form.

## 3. Can you provide an example of binomial expansion?

For example, to expand (x+2)^3, we would use the binomial theorem formula: (n choose k) * x^(n-k) * 2^k. Substituting in our values, we get (3 choose 0) * x^(3-0) * 2^0 + (3 choose 1) * x^(3-1) * 2^1 + (3 choose 2) * x^(3-2) * 2^2 + (3 choose 3) * x^(3-3) * 2^3. Simplifying this, we get x^3 + 3x^2 * 2 + 3x * 4 + 8, which is the expanded form of (x+2)^3.

## 4. What is the purpose of binomial expansion?

Binomial expansion is used to simplify and solve complex mathematical expressions, particularly those involving polynomials. It allows us to find the coefficients and terms in an expanded expression, making it easier to work with and solve equations.

## 5. Are there any limitations to binomial expansion?

Binomial expansion can only be used for binomial expressions, which have two terms. It also relies on the binomial theorem, which only works for whole number powers. Additionally, as the power increases, the number of terms in the expanded expression also increases, making it more difficult to calculate by hand.

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