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The binomial theorem is a powerful tool in algebra for expanding expressions raised to a power. It involves a formula that expands the power of a sum in terms of the sum's individual terms. For example, when we have an expression like \( (a + b)^n \), where \( n \) is a non-negative integer, the theorem tells us we can expand this into a sum involving powers of \( a \) and \( b \) multiplied by binomial coefficients. In practical terms, it allows us to write out the complete expanded form of polynomials like \( (x^{2} - 2)^3 \) without having to multiply the binomial out the long way, term by term.

Applying this to the exercise, we see that each term in the expansion is a product of a binomial coefficient and the terms \( a \) and \( b \) raised to specific powers that collectively add up to \( n \). The theorem not only makes it easier to expand expressions but also helps in understanding the symmetrical nature of the coefficients in such expansions.

The binomial coefficient, often read as 'n choose k', is a fundamental part of the binomial theorem. It represents the number of ways to choose \( k \) elements from a set of \( n \) distinct elements, and it's denoted as \( {n \choose k} \). The mathematics behind it stems from combinatorics, and it is defined by the formula \( \frac{n!}{k!(n-k)!} \), where \( n! \) and \( k! \) are factorial values.

In the given exercise, calculating the binomial coefficients requires us to plug in the values for \( n \) and \( k \) into the formula. For instance, \( {3 \choose 1} \) corresponds to the number of ways to choose one item from a set of three, which is simply three. Understanding binomial coefficients is crucial as they weigh each term in the polynomial expansion and hints at the probability and combinations aspect of mathematics.

Polynomial expansion is the process of expressing a polynomial that is raised to a power as a sum of terms with different powers. This is important in solving algebraic problems where direct multiplication is cumbersome. The binomial theorem is a specific case of polynomial expansion when the polynomial has exactly two terms.

For the exercise \( (x^{2}-2)^{3} \) we see that the expansion leads to a polynomial of higher degree. Each term in the expansion of \( (a+b)^n \) has a certain structure: the sum of the exponents in each term is \( n \) which reflects the original power the binomial was raised to. The binomial theorem gives us a formulaic approach to generate these terms without actually multiplying out the entire expression, providing a shortcut to polynomial expansion.

The factorial function is defined as the product of all positive integers up to a given number. It's represented by an exclamation point \( n! \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). This function is fundamental in various branches of mathematics, including algebra, number theory, and especially combinatorics.

In the context of our exercise, factorials are used to calculate the binomial coefficients — a key part of the binomial theorem. Remember, the factorial of zero \( (0!) \) is defined as 1, which is an important convention when interpreting the \( {n \choose k} \) formula. Calculating factorials is straightforward for small numbers, but for larger numbers, it's often better to use simplification techniques or factor cancellation when plugging them into the binomial coefficient formula.