Use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³

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Recall the Binomial Theorem formula for expanding \((a + b)^n\): \[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \] where \(\binom{n}{k}\) is the binomial coefficient.

Identify the values in the given expression \((x + 2)^3\): here, \(a = x\), \(b = 2\), and \(n = 3\).

Write out each term of the expansion using the formula: \[ \binom{3}{0} x^{3} 2^{0} + \binom{3}{1} x^{2} 2^{1} + \binom{3}{2} x^{1} 2^{2} + \binom{3}{3} x^{0} 2^{3} \]

Calculate each binomial coefficient \(\binom{3}{k}\) using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!} \) or from Pascal's triangle.

Simplify each term by evaluating the powers and coefficients, then combine all terms to write the final expanded expression.

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주요 개념

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Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It uses binomial coefficients, often represented by combinations, to determine the coefficients of each term in the expansion.

Binomial Coefficients

Binomial coefficients are the numerical factors in the expansion of (a + b)^n, calculated using combinations: C(n, k) = n! / (k!(n-k)!). They represent the number of ways to choose k elements from n and determine the coefficients of each term.

Simplifying Polynomial Expressions

After expanding a binomial using the theorem, simplifying involves combining like terms and performing arithmetic operations to write the expression in its simplest polynomial form, making it easier to interpret and use.