Binomial Expansion

A binomial is an algebraic expression with exactly two terms joined by a plus or minus sign. Examples:

• \(a + b\)
• \(x - 3\)
• \(2p + 5q\)

Binomial expansion is about finding the full expression when a binomial is raised to a power, like \((a + b)^2\), \((x - 3)^4\), or in general \((a + b)^n\).

To understand \((a + b)^n\), think of it as multiplying \((a + b)\) by itself again and again.

For small powers:

• \((a + b)^1 = a + b\)
• \((a + b)^2 = (a + b)(a + b)\)
• \((a + b)^3 = (a + b)(a + b)(a + b)\)

When expanded, each term comes from choosing either \(a\) or \(b\) from every bracket, and then multiplying all the choices together. The binomial theorem gives a shortcut formula so there is no need to multiply everything by hand every time.

In the expansion of \((a + b)^n\), each term has a numerical factor in front. These numbers are called binomial coefficients.

For example:

• \((a + b)^2 = a^2 + 2ab + b^2\) → coefficients are 1, 2, 1
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) → coefficients are 1, 3, 3, 1

These coefficients can be written using combinations (nCr):

\(\binom{n}{0},\ \binom{n}{1},\ \binom{n}{2},\ldots,\ \binom{n}{n}\)

Here, \(\binom{n}{r}\) is read as "n choose r" and counts how many ways to choose \(r\) objects out of \(n\).

Pascal's triangle is a simple way to list binomial coefficients without using formulas.

Start with 1 at the top, then build each new row so that every inner number is the sum of the two numbers just above it. The first few rows are:

          1
        1   1
      1   2   1
    1   3   3   1
  1   4   6   4   1

Each row gives the coefficients for a binomial expansion:

• Row 0: \((a + b)^0\)
• Row 1: \((a + b)^1\)
• Row 2: \((a + b)^2\)
• Row 3: \((a + b)^3\)
• Row 4: \((a + b)^4\)

For example, row 3 is 1, 3, 3, 1 and matches the coefficients in \((a + b)^3\).

The binomial theorem gives a general formula for expanding \((a + b)^n\):

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

This means:

• Start with \(r = 0\) and go up to \(r = n\).
• Each term has coefficient \(\binom{n}{r}\).
• The power of \(a\) is \(n - r\).
• The power of \(b\) is \(r\).

So the expansion looks like:

\[(a + b)^n = \binom{n}{0} a^{n} b^{0} + \binom{n}{1} a^{n-1} b^{1} + \binom{n}{2} a^{n-2} b^{2} + \cdots + \binom{n}{n} a^{0} b^{n}.\]

Example 1: Expand \((x + y)^3\)

Using the pattern from Pascal's triangle (1, 3, 3, 1):

\[(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\]

Example 2: Expand \((2a - b)^4\)

Coefficients for power 4 are 1, 4, 6, 4, 1.

Write each term carefully:

• 1st term: \(\binom{4}{0} (2a)^4 (-b)^0 = 16a^4\)
• 2nd term: \(\binom{4}{1} (2a)^3 (-b)^1 = 4 \cdot 8a^3(-b) = -32a^3b\)
• 3rd term: \(\binom{4}{2} (2a)^2 (-b)^2 = 6 \cdot 4a^2 b^2 = 24a^2b^2\)
• 4th term: \(\binom{4}{3} (2a)^1 (-b)^3 = 4 \cdot 2a(-b^3) = -8ab^3\)
• 5th term: \(\binom{4}{4} (2a)^0 (-b)^4 = b^4\)

So,

\[(2a - b)^4 = 16a^4 - 32a^3b + 24a^2b^2 - 8ab^3 + b^4.\]