Sequences, Series and Binomial Theorem
Binomial Theorem
Learning Objectives
By the end of this section, you will be able to:
- Use Pascal’s Triangle to expand a binomial
- Evaluate a binomial coefficient
- Use the Binomial Theorem to expand a binomial
Before you get started, take this readiness quiz.
Use Pascal’s Triangle to Expand a Binomial
In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded
To expand we recognize that this is and multiply.
To find a method that is less tedious that will work for higher expansions like we again look for patterns in some expansions.
| Number of terms | First term | Last term | |
|---|---|---|---|
| 2 | |||
| 3 | |||
| 4 | |||
| 5 | |||
| 6 | |||
| n |
Notice the first and last terms show only one variable. Recall that so we could rewrite the first and last terms to include both variables. For example, we could expand to show each term with both variables.
Generally, we don’t show the zero exponents, just as we usually write x rather than 1x.
- The number of terms is
- The first term is and the last term is
- The exponents on a decrease by one on each term going left to right.
- The exponents on b increase by one on each term going left to right.
- The sum of the exponents on any term is n.
Let’s look at an example to highlight the last three patterns.
From the patterns we identified, we see the variables in the expansion of would be
To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.
The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
This triangle gives the coefficients of the terms when we expand binomials.
In the next example, we will use this triangle and the patterns we recognized to expand the binomial.
Use Pascal’s Triangle to expand
We know the variables for this expansion will follow the pattern we identified. The nonzero exponents of x will start at six and decrease to one. The nonzero exponents of y will start at one and increase to six. The sum of the exponents in each term will be six. In our pattern, and
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