Algebraic Formulas, Expansions, and the Binomial Theorem

$$(x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^k a^{n-k}$$ binomial theorem, which states that for any non-negative integer $n$, the expansion of $(x+a)^n$ is given by the sum of the terms involving $x$ and $a$, each raised to some power:
Here: $n$ müsbət tam ədəd.
$n$ is a non-negative integer.
$\sum$ represents the summation symbol, which means to sum the terms generated by the formula inside the parentheses for $k = 0$ to $n$.

$\binom{n}{k}$ or $_nC_k$ (also written as C(n,k) or "n choose k") represents the binomial coefficient, which is the number of ways to choose $k$ items from a set of $n$ items. It is calculated using the formula: $$\binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where "$!$" denotes the factorial function.

$x^k$ and $a^{n-k}$ are the terms involving $x$ and $a$, each raised to some power.
In the expansion of $(x+a)^n$, there are $(n+1)$ terms, with each term being a product of the binomial coefficient, $x$ raised to some power, and $a$ raised to some other power. The powers of $x$ and $a$ decrease and increase, respectively, as we move from the first term to the last term in the expansion.

For example, $(x+a)^3$: $$(x+a)^3 = (nC0) x^3 a^0+ (nC1) x^2 a^1+ (nC2) x^1 a^2+ (nC3) x^0 a^3$$ Using the binomial coefficients: $$(x+a)^3 = 1(x^3 )(a^0)+3(x^2 )(a^1 )+3(x^1 )(a^2 )+ 1(x^0 )(a^3 )= x^3+3x^2 a+3xa^2+a^3$$

This formula allows us to easily expand binomials raised to any power without having to manually apply the distributive property multiple times.