Sequences, Probability, and Counting Theory

# Binomial Theorem

### Learning Objectives

In this section, you will:

• Apply the Binomial Theorem.

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to findwithout multiplying the binomial by itself times.

### Identifying Binomial Coefficients

In Counting Principles, we studied combinations. In the shortcut to findingwe will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation instead of but it can be calculated in the same way. So

The combinationis called a binomial coefficient. An example of a binomial coefficient is

### Binomial Coefficients

If and are integers greater than or equal to 0 with then the binomial coefficient is

Is a binomial coefficient always a whole number?

Yes. Just as the number of combinations must always be a whole number, a binomial coefficient will always be a whole number.

### Finding Binomial Coefficients

Find each binomial coefficient.

[reveal-answer q=”fs-id1165137933188″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137933188″]Use the formula to calculate each binomial coefficient. You can also use the function on your calculator.

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#### Analysis

Notice that we obtained the same result for parts (b) and (c). If you look closely at the solution for these two parts, you will see that you end up with the same two factorials in the denominator, but the order is reversed, just as with combinations.

### Try It

Find each binomial coefficient.

[reveal-answer q=”fs-id1165137653724″]Show Solution[/reveal-answer]
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1. 35
2. 330

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### Using the Binomial Theorem

When we expand by multiplying, the result is called a binomial expansion, and it includes binomial coefficients. If we wanted to expand we might multiply by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

First, letâ€™s examine the exponents. With each successive term, the exponent for decreases and the exponent for increases. The sum of the two exponents is for each term.

Next, letâ€™s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:

These patterns lead us to the Binomial Theorem, which can be used to expand any binomial.

Another way to see the coefficients is to examine the expansion of a binomial in general form,to successive powers 1, 2, 3, and 4.

Can you guess the next expansion for the binomial

See (Figure), which illustrates the following:

• There are terms in the expansion of
• The degree (or sum of the exponents) for each term is
• The powers on begin with and decrease to 0.
• The powers on begin with 0 and increase to
• The coefficients are symmetric.

To determine the expansion on we see thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of the pattern is as follows:

• Introduce and then for each successive term reduce the exponent on by 1 until is reached.
• Introduce and then increase the exponent on by 1 until is reached.

The next expansion would be

But where do those coefficients come from? The binomial coefficients are symmetric. We can see these coefficients in an array known as Pascal’s Triangle, shown in (Figure).

To generate Pascalâ€™s Triangle, we start by writing a 1. In the row below, row 2, we write two 1â€™s. In the 3rd row, flank the ends of the rows with 1â€™s, and add to find the middle number, 2. In the row, flank the ends of the row with 1â€™s. Each element in the triangle is the sum of the two elements immediately above it.

To see the connection between Pascalâ€™s Triangle and binomial coefficients, let us revisit the expansion of the binomials in general form.

### The Binomial Theorem

The Binomial Theorem is a formula that can be used to expand any binomial.

### How To

Given a binomial, write it in expanded form.

1. Determine the value of according to the exponent.
2. Evaluate the through using the Binomial Theorem formula.
3. Simplify.

### Expanding a Binomial

Write in expanded form.

[reveal-answer q=”221236″]Show Solution[/reveal-answer]
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1. Substitute into the formula. Evaluate the through terms. Simplify.
2. Substitute into the formula. Evaluate the through terms. Notice that is in the place that was occupied by and that is in the place that was occupied by So we substitute them. Simplify.

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#### Analysis

Notice the alternating signs in part b. This happens becauseraised to odd powers is negative, butraised to even powers is positive. This will occur whenever the binomial contains a subtraction sign.

### Try It

Write in expanded form.

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### Using the Binomial Theorem to Find a Single Term

Expanding a binomial with a high exponent such ascan be a lengthy process.

Sometimes we are interested only in a certain term of a binomial expansion. We do not need to fully expand a binomial to find a single specific term.

Note the pattern of coefficients in the expansion of

The second term isThe third term isWe can generalize this result.

### The (r+1)th Term of a Binomial Expansion

Theterm of the binomial expansion ofis:

### How To

Given a binomial, write a specific term without fully expanding.

1. Determine the value of according to the exponent.
2. Determine
3. Determine
4. Replace in the formula for the term of the binomial expansion.

### Writing a Given Term of a Binomial Expansion

Find the tenth term ofwithout fully expanding the binomial.

[reveal-answer q=”fs-id1165137834759″]Show Solution[/reveal-answer]
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Because we are looking for the tenth term, we will use in our calculations.

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### Try It

Find the sixth term ofwithout fully expanding the binomial.

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Access these online resources for additional instruction and practice with binomial expansion.

### Key Equations

 Binomial Theorem term of a binomial expansion

### Key Concepts

• is called a binomial coefficient and is equal to See (Figure).
• The Binomial Theorem allows us to expand binomials without multiplying. See (Figure).
• We can find a given term of a binomial expansion without fully expanding the binomial. See (Figure).

### Section Exercises

#### Verbal

What is a binomial coefficient, and how it is calculated?

[reveal-answer q=”fs-id1165135501149″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135501149″]A binomial coefficient is an alternative way of denoting the combination It is defined as[/hidden-answer]

What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

What is the Binomial Theorem and what is its use?

[reveal-answer q=”fs-id1165137452921″]Show Solution[/reveal-answer]
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The Binomial Theorem is defined asand can be used to expand any binomial.

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When is it an advantage to use the Binomial Theorem? Explain.

#### Algebraic

For the following exercises, evaluate the binomial coefficient.

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15

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[reveal-answer q=”fs-id1165135484154″]Show Solution[/reveal-answer]
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35

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10

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[reveal-answer q=”207307″]Show Solution[/reveal-answer]
[hidden-answer a=”207307″]12,376[/hidden-answer]

For the following exercises, use the Binomial Theorem to expand each binomial.

[reveal-answer q=”fs-id1165137831242″]Show Solution[/reveal-answer]
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For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

[reveal-answer q=”fs-id1165137527392″]Show Solution[/reveal-answer]
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For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fourth term of

The fourth term of

[reveal-answer q=”fs-id1165137827785″]Show Solution[/reveal-answer]
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The third term of

The eighth term of

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The seventh term of

The fifth term of

[reveal-answer q=”fs-id1165137673496″]Show Solution[/reveal-answer]
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The tenth term of

The ninth term of

[reveal-answer q=”fs-id1165137423625″]Show Solution[/reveal-answer]
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The fourth term of

The eighth term of

[reveal-answer q=”fs-id1165137692780″]Show Solution[/reveal-answer]
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#### Graphical

For the following exercises, use the Binomial Theorem to expand the binomial Then find and graph each indicated sum on one set of axes.

Find and graphsuch thatis the first term of the expansion.

Find and graphsuch thatis the sum of the first two terms of the expansion.

[reveal-answer q=”fs-id1165137575802″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137575802″]

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Find and graphsuch that is the sum of the first three terms of the expansion.

Find and graphsuch thatis the sum of the first four terms of the expansion.

[reveal-answer q=”fs-id1165135408467″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135408467″]

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Find and graphsuch thatis the sum of the first five terms of the expansion.

#### Extensions

In the expansion ofeach term has the formsuccessively takes on the valueIfwhat is the corresponding term?

[reveal-answer q=”fs-id1165137871611″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137871611″]

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In the expansion ofthe coefficient ofis the same as the coefficient of which other term?

Consider the expansion ofWhat is the exponent of in the term?

[reveal-answer q=”fs-id1165137731710″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137731710″]

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Findand write the answer as a binomial coefficient in the formProve it. Hint: Use the fact that, for any integersuch that

[reveal-answer q=”fs-id1165137692126″]Show Solution[/reveal-answer]
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Proof:

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\begin{array}{}\\ \\ \\ \,\,\,\,\,\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-\left(k-1\right)\right)!}\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!}{\left(n-k+1\right)k!\left(n-k\right)!}+\frac{kn!}{k\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!+kn!}{k!\left(n-k+1\right)!}\\ =\frac{\left(n+1\right)n!}{k!\left(\left(n+1\right)-k\right)!}\\ =\frac{\left(n+1\right)!}{k!\left(\left(n+1\right)-k\right)!}\\ =\left(\begin{array}{c}n+1\\ k\end{array}\right)\end{array}

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Which expression cannot be expanded using the Binomial Theorem? Explain.

[reveal-answer q=”fs-id1165135176542″]Show Solution[/reveal-answer]
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The expressioncannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

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### Glossary

binomial coefficient
the number of ways to choose r objects from n objects where order does not matter; equivalent todenoted
binomial expansion
the result of expandingby multiplying
Binomial Theorem
a formula that can be used to expand any binomial

## License

Algebra and Trigonometry by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.