L23-L24: Binomial Expansion

Lecturer: EN. Hisham Safuan Mohamad Sukri

Learning Outcomes

By the end of this lecture and the associated tutorial, students should be able to:

  • Compute and simplify factorial expressions
  • Construct Pascal's triangle to identify binomial coefficients
  • Use the standard theorem to expand $(a+b)^n$ for $n \in \mathbb{Z}^+$
  • Find a specific term without expanding the entire expression
  • Apply the generalized binomial theorem for cases when $n$ is a non-positive integer (negative integer, fraction, etc.) and determine the validity range
  • Use the binomial expansion to approximate complex values in decimal form

Factorial

The $n!$ notation:

  • We read $n!$ as "$n$ factorial"
  • $n!$ is the product of all positive integers from 1 up to a given non-negative integer
  • In general, $n!$ is defined as: $$n! = n \times (n-1) \times (n-2) \times \cdots \times 3 \times 2 \times 1$$ with $n \in \mathbb{N}$ and $0! = 1$
  • Factorial forms a recursive property: $$(n+1)! = (n+1) \times n! = (n+1) \times n \times (n-1)!$$

Example 1 Evaluate: a) $6!$ b) $\dfrac{8!}{6!}$ c) $\dfrac{4!}{(6-4)!}$

Example 2

  1. Prove that $(n+2) \times n! = n! + (n+1)!$
  2. If $(n+1)! = 12 \times (n-1)!$, find the value of $n$

Combinations (Binomial Coefficients)

The notation ${}^nC_r$ or $\displaystyle\binom{n}{r}$ is called the binomial coefficient as it is found in binomial expansion.

The binomial coefficient is defined as: $${}^nC_r = \binom{n}{r} = \frac{n!}{(n-r)! , r!}$$ where $n$ is a positive integer, $r \in \mathbb{N}$ and $0 \le r \le n$.

Special cases: $${}^nC_0 = \binom{n}{0} = \frac{n!}{n! , 0!} = 1$$ $${}^nC_1 = \binom{n}{1} = \frac{n!}{(n-1)! , 1!} = n$$ $${}^nC_2 = \binom{n}{2} = \frac{n(n-1)}{2!}$$

General form (useful for computation): $${}^nC_r = \binom{n}{r} = \frac{n \times (n-1) \times (n-2) \times \cdots \times (n-r+1)}{r!}, \quad n \ge r$$

Example 3 Show that: $$\binom{n}{r-1} + \binom{n}{r} = \binom{n+1}{r}$$

Binomial Expansions — The Binomial Theorem

A binomial is a sum of two terms $a+b$, which is usually an algebraic expression. The $a$ and $b$ can also represent specific numbers.

If $n$ is a positive integer, then a general formula for expanding $(a+b)^n$ is given by the Binomial Theorem:

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

Expanded form: $$(a+b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n-1}ab^{n-1} + \binom{n}{n}b^n$$

Characteristics of the expansion $(a+b)^n$

  1. There are a total of $(n+1)$ terms in the series
  2. The first term and last term are $a^n$ and $b^n$, respectively
  3. The power of $a$ decreases by one from left to right, while the power of $b$ increases by one; the total power adds up to $n$ for each term
  4. The coefficients of the terms are symmetrical and follow the pattern in Pascal's Triangle

Pascal's Triangle

        1
      1   1
    1   2   1
  1   3   3   1
1   4   6   4   1
1  5  10  10  5  1

Example 4 Expand: a) $(x+2y)^3$ b) $(2x-3y)^4$

Example 5 Use the expansion $(a+b)^6$ to approximate $(2.02)^6$ correct to four decimal places.

Example 6 a) Find the fifth term in the expansion of $\left(x^3 + \sqrt{y}\right)^{13}$ b) Find the coefficient of $x^2$ in the expansion of $\left(\dfrac{3}{x} - 5x^3\right)^6$ c) Expand $(1+2x-x^2)^5$ in ascending powers of $x$ up to the term in $x^3$

General Binomial Theorem (Non-Positive Integer Powers)

The binomial expansion when $n$ is a non-positive integer (or more generally, not a positive integer — including negative integers and fractions) takes the form $(1+x)^n$ where $|x| < 1$.

  • The expansion is an infinite series
  • This series expansion is called the binomial series and is only valid for $|x| < 1$:

$$(1+x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \cdots, \qquad |x| < 1$$

Differences Between the Two Cases

When $n$ is positive integer When $n$ is non-positive integer
The series $(1+x)^n = 1 + {}^nC_1 x + {}^nC_2 x^2 + {}^nC_3 x^3 + \cdots + x^n$ The series $(1+x)^n = 1 + nx + \dfrac{n(n-1)}{2}x^2 + \dfrac{n(n-1)(n-2)}{3!}x^3 + \cdots$
The series is finite, ending at $x^n$; valid for all $x \in \mathbb{R}$ The series is infinite and converges when $
If $x$ does not satisfy the validity range, the series diverges

Example 7 Expand each of the following functions as an ascending series in $x$, up to and including the term in $x^3$. State the range of $x$ such that the expansion is valid. a) $\sqrt{1+2x}$ b) $(1+x)^{-4}$

Example 8 Expand $(1+x)^{\frac{1}{3}}$ in ascending powers of $x$ up to the term in $x^2$. Hence, evaluate $\sqrt[3]{8.064}$ correct to four decimal places.

Example 9 Express $f(x) = \dfrac{17-x}{(2-x)(3+x)}$ in partial fractions.

Hence, expand $f(x)$ as a series in ascending powers of $x$ up to the term in $x^2$. Find the range of values of $x$ for which the expansion is valid.

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