Apply the binomial theorem

The binomial theorem represents $(a+b)^n$ , for a nonnegative integer $n$ , as $\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$ , enabling accurate expansion, identification of particular terms or coefficients, and interpretation of $\binom{n}{k}$ as the number of ways to choose which factors contribute $b$ . It connects algebraic expansion with combinations and binomial probability; noninteger or negative exponents and infinite-series generalizations are outside this scope.

Detailed Explanation: Apply the binomial theorem

To expand a power of a binomial, use the binomial theorem:

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

Here, $\binom{n}{k}$ gives the coefficient, and $k$ tells how many factors contribute $b$ .

Example: Expand $(2x-3)^4$

Identify the parts:

$a=2x$
$b=-3$
$n=4$

Substitute into the theorem:

(2x-3)^4 =\binom{4}{0}(2x)^4 +\binom{4}{1}(2x)^3(-3) +\binom{4}{2}(2x)^2(-3)^2 +\binom{4}{3}(2x)(-3)^3 +\binom{4}{4}(-3)^4

Use the binomial coefficients:

\binom{4}{0}=1,\quad \binom{4}{1}=4,\quad \binom{4}{2}=6,\quad \binom{4}{3}=4,\quad \binom{4}{4}=1

Now simplify each term:

\begin{aligned} (2x-3)^4 &=1(2x)^4+4(2x)^3(-3)+6(2x)^2(-3)^2\\ &\quad+4(2x)(-3)^3+1(-3)^4\\ &=16x^4-96x^3+216x^2-216x+81 \end{aligned}

Therefore,

\boxed{(2x-3)^4=16x^4-96x^3+216x^2-216x+81}

The signs alternate here because $b=-3$ , so odd powers of $b$ are negative.

Learn by doing: Apply the binomial theorem

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Binomial Theorem - Polynomial with Two Integers and Triangle to Expanded Polynomial

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