Question

What is the binomial theorem with example?

Answer 1

The Binomial Theorem provides a formula to expand expressions of the form \((a + b)^n\), where \(a\), \(b\), and \(n\) are constants, and \(n\) is a non-negative integer. It allows us to expand the power of a binomial without directly multiplying it out repeatedly.

Binomial Theorem Formula:

The binomial expansion of \((a + b)^n\) is given by:

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

Where:

1. \(\binom{n}{k}\) is the binomial coefficient, and it is calculated as:
   

  \[
  \binom{n}{k} = \frac{n!}{k!(n-k)!}
  \]
  (This gives the number of ways to choose \(k\) items from a set of \(n\) items.)

1. The summation \(\sum_{k=0}^{n}\) indicates that we will add up terms from \(k = 0\) to \(k = n\).
   

Example: Expanding \((x + 2)^3\)

Let’s expand \((x + 2)^3\) using the binomial theorem.

Here, \(a = x\), \(b = 2\), and \(n = 3\).

We apply the binomial formula:

\[
(x + 2)^3 = \sum_{k=0}^{3} \binom{3}{k} x^{3-k} 2^k
\]

Let’s break it down term by term:

1. When \(k = 0\):
   

  \[
  \binom{3}{0} x^{3-0} 2^0 = 1 \cdot x^3 \cdot 1 = x^3
  \]

1. When \(k = 1\):
   

  \[
  \binom{3}{1} x^{3-1} 2^1 = 3 \cdot x^2 \cdot 2 = 6x^2
  \]

1. When \(k = 2\):
   

  \[
  \binom{3}{2} x^{3-2} 2^2 = 3 \cdot x \cdot 4 = 12x
  \]

1. When \(k = 3\):
   

  \[
  \binom{3}{3} x^{3-3} 2^3 = 1 \cdot 1 \cdot 8 = 8
  \]

Putting it all together:

\[
(x + 2)^3 = x^3 + 6x^2 + 12x + 8
\]

This is the expanded form of \((x + 2)^3\).

Key Points to Remember:

1. The binomial theorem helps expand binomials raised to any power.
   

1. Each term in the expansion involves the binomial coefficient, powers of \(a\) and \(b\), and the appropriate terms from the summation.
   

1. The binomial coefficient \(\binom{n}{k}\) can be computed from factorials, and it gives the number of ways to choose \(k\) items from \(n\).
   

Is there a specific example or concept within the binomial theorem that you’d like to dive deeper into?