Write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)⁸

Verified step by step guidance

1. Recall the Binomial Theorem, which states that for any positive integer \(n\), the expansion of \((a + b)^n\) is given by: \[ (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \] where \(\binom{n}{k}\) is the binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).
2. Identify the values of \(a\), \(b\), and \(n\) in the expression \((x + 2)^8\). Here, \(a = x\), \(b = 2\), and \(n = 8\).
3. Write the first three terms of the expansion by substituting \(k = 0, 1, 2\) into the binomial formula: - For \(k=0\): \[ \binom{8}{0} x^{8-0} 2^0 = \binom{8}{0} x^8 \cdot 1 \] - For \(k=1\): \[ \binom{8}{1} x^{8-1} 2^1 = \binom{8}{1} x^7 \cdot 2 \] - For \(k=2\): \[ \binom{8}{2} x^{8-2} 2^2 = \binom{8}{2} x^6 \cdot 4 \]
4. Calculate the binomial coefficients \(\binom{8}{0}\), \(\binom{8}{1}\), and \(\binom{8}{2}\) using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) or Pascal's Triangle.
5. Multiply the coefficients by the powers of \(x\) and \(2\), then simplify each term to write the first three terms of the expansion in simplified form.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n. It states that the expansion is the sum of terms involving binomial coefficients multiplied by powers of a and b. This theorem allows us to find any term in the expansion without fully multiplying the expression.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term in the expansion.

Simplifying Powers and Terms

After applying the binomial theorem, each term involves powers of the variables and constants. Simplifying these powers and multiplying constants correctly is essential to express the terms in their simplest form. This step ensures the final expansion is clear and easy to interpret.