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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.
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.
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.
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.
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.