- Apply the Binomial Theorem
Binomial Coefficients
The Main Idea
- Binomial Definition: A polynomial with two terms is called a binomial.
- Binomial Theorem: A powerful tool for expanding expressions raised to a power, useful in combination problems.
- Binomial Coefficients: The coefficients in the expansion of [latex](x+y)^n[/latex], following a pattern visible in Pascal’s Triangle.
- Binomial Coefficient Formula: For integers [latex]n \geq r \geq 0[/latex], [latex]\binom{n}{r} = C(n,r) = \frac{n!}{r!(n-r)!}[/latex]
- Symmetry Property: [latex]\binom{n}{r} = \binom{n}{n-r}[/latex]
- [latex]\left(\begin{gathered}7\\ 3\end{gathered}\right)[/latex]
- [latex]\left(\begin{gathered}11\\ 4\end{gathered}\right)[/latex]
The Binomial Theorem
The Main Idea
- Binomial Expansion: The result of expanding [latex](x+y)^n[/latex].
- Pascal’s Triangle: A triangular array of binomial coefficients that follows a simple rule of construction.
- Binomial Theorem: A formula for expanding any binomial to any power without performing repeated multiplication.
Patterns in Binomial Expansions
When expanding [latex](x+y)^n[/latex]:
- There are [latex]n+1[/latex] terms in the expansion.
- The degree (sum of exponents) for each term is [latex]n[/latex].
- Powers of [latex]x[/latex] decrease from [latex]n[/latex] to [latex]0[/latex].
- Powers of [latex]y[/latex] increase from [latex]0[/latex] to [latex]n[/latex].
- Coefficients are symmetric.
Pascal’s Triangle and Binomial Coefficients
Pascal’s Triangle is formed by:
- Starting with 1 at the top
- Each number below is the sum of the two numbers above it
- Rows are symmetric
The numbers in Pascal’s Triangle correspond to the coefficients in binomial expansions.
The Binomial Theorem
[latex](x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k[/latex]
Or written out:
[latex](x+y)^n = \binom{n}{0}x^n + \binom{n}{1}x^{n-1}y + \binom{n}{2}x^{n-2}y^2 + ... + \binom{n}{n-1}xy^{n-1} + \binom{n}{n}y^n[/latex]
Where [latex]\binom{n}{k}[/latex] is the binomial coefficient.
- [latex]{\left(x-y\right)}^{5}[/latex]
- [latex]{\left(2x+5y\right)}^{3}[/latex]
You can view the transcript for “Ex 2: The Binomial Theorem Using Pascal’s Triangle” here (opens in new window).
You can view the transcript for “The Binomial Theorem using Combination” here (opens in new window).
You can view the transcript for “Ex 1: The Binomial Theorem Using Combinations” here (opens in new window).
Using the Binomial Theorem to Find a Single Term
The Main Idea
The Binomial Theorem can be used to find a specific term in a binomial expansion without expanding the entire expression. This is particularly useful for binomials with high exponents.
Key Formula
The [latex](r+1)[/latex]th term of the binomial expansion of [latex](x+y)^n[/latex] is:
[latex]\binom{n}{r} x^{n-r} y^r[/latex]
Where:
- [latex]n[/latex] is the exponent of the binomial
- [latex]r[/latex] is one less than the position of the term we’re looking for
How to Find a Specific Term
- Identify the exponent [latex]n[/latex] of the binomial.
- Determine which term you’re looking for (let’s call it the [latex]k[/latex]th term).
- Calculate [latex]r = k - 1[/latex].
- Apply the formula: [latex]\binom{n}{r} x^{n-r} y^r[/latex].