89 Binomial Theorem

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.

Patterns in the expansion of

• 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

In the next example we want to expand a binomial with one variable and one constant. We need to identify the a and b to carefully apply the pattern.

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\begin{array}{c}{\left(a+b\right)}^{n}={a}^{n}+___{a}^{n-1}{b}^{1}+___{a}^{n-2}{b}^{2}+...+___{a}^{1}{b}^{n-1}+{b}^{n}\\ {\left(x+3\right)}^{5}={x}^{5}+___{x}^{4}·{3}^{1}+___{x}^{3}·{3}^{2}+___{x}^{2}·{3}^{3}+___{x}^{1}·{3}^{4}+{3}^{5}\end{array}

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In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the a and b of the pattern, we must once again carefully apply the pattern.

Use Pascal’s Triangle to expand

We identify the a and b of the pattern.

In our pattern, and

Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.

To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation which is called a binomial coefficient. We read as “n choose r” or “n taken r at a time”.

Binomial Coefficient

A binomial coefficient where r and n are integers with is defined as

We read as “n choose r” or “n taken r at a time”.

In the previous example, parts (a), (b), (c) demonstrate some special properties of binomial coefficients.

Properties of Binomial Coefficients

Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.

Binomial Theorem

For any real numbers a and b, and positive integer n,

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents.

Evaluate the coefficients. Remember,

Notice that when we expanded in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle.

The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents and evaluate the coefficients. Remember,

Things can get messy when both terms have a coefficient and a variable.

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents.

Evaluate the coefficients. Remember,

The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.

Notice, that in each case the exponent on the b is one less than the number of the term. The term is the term where the exponent of b is r. So we can use the format of the term to find the value of a specific term.

Find a Specific Term in a Binomial Expansion

The term in the expansion of is

Find the fourth term of

In our pattern, and
We are looking for the fourth term.
Write the formula.
Substitute in the values, and
Simplify.
Simplify.

Find the coefficient of the term of

In our pattern, then and
We are looking for the coefficient of the term.     Since and we know
Write the formula.
Substitute in the values, and
Simplify.
Simplify.
Simplify.
	The coefficient of the term is 2268.