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When expanding expressions such as \((x + \sqrt{2})^8\), binomial coefficients play a vital role. These coefficients are part of the Binomial Theorem, a fundamental concept in algebra. They determine the weight of each term in the polynomial expansion.
The binomial coefficient, denoted by \(\binom{n}{k}\), is a way to choose \(k\) elements from a set of \(n\) elements without regard to the order of selection. The formula for calculating a binomial coefficient is:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Here, \(n!\) (n factorial) is the product of all positive integers up to \(n\). These coefficients essentially determine how many times each term appears in the expansion.
In the example provided, for \((x + \sqrt{2})^8\), the calculated binomial coefficients for each term from \(k = 0\) to \(k = 8\) are: 1, 8, 28, 56, 70, 56, 28, 8, 1. These numbers are crucial for determining the individual terms in our expanded expression.

Polynomial expansion involves expressing a power of a binomial as a sum of terms. Each term is a product of a binomial coefficient, a power of the first term, and a power of the second term.
Using the Binomial Theorem, the polynomial expansion of \((a + b)^n\) can be expressed as:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

For instance, expanding \((x + \sqrt{2})^8\) involves identifying \(a = x\) and \(b = \sqrt{2}\), and substituting into the formula:

• \((x+\sqrt{2})^8 = \sum_{k=0}^{8} \binom{8}{k} x^{8-k} (\sqrt{2})^k\)

Each term is computed by varying \(k\) from 0 to 8, evaluating the powers of \(x\) and \(\sqrt{2}\), and applying the corresponding binomial coefficient. This results in a polynomial with multiple terms, each term representing a different component of the expansion.

Algebraic manipulation is the process of simplifying and rearranging expressions. During polynomial expansion using the binomial theorem, we need to manipulate the terms to simplify the final expression.
In the example \((x + \sqrt{2})^8\), each term calculated includes a coefficient, a power of \(x\), and a power of \(\sqrt{2}\). For instance, one term is calculated as \(8 \times x^7 \times \sqrt{2} = 8\sqrt{2} x^7\).
Simplification requires careful multiplication and understanding of exponent rules, especially when dealing with square roots or higher powers. The expression must be consistently rearranged from terms involving the highest power of \(x\) down to the constant term.

• The term \(56 \times x^5 \times 2\sqrt{2}\) simplifies to \(112\sqrt{2} x^5\).
• Each term's simplification might involve a numerical factor and a factor involving \(\sqrt{2}\).

Ultimately, all terms are combined to form the fully expanded and simplified polynomial expression. This manipulation ensures clarity and correctness in expressing the polynomial.