In this explainer, we will learn how to expand any binomial expression of the form using combinations.
We can expand a binomial expression raised to the second power as
We can use this expression to expand a binomial expression raised to the third power as
While we can recursively continue to write the expansion of higher powers of a binomial expression, it would be quite time consuming to obtain, for instance, even the 10th power of a binomial expression using this approach.
The binomial theorem gives a direct formula for an expansion of an arbitrary positive integer power of a binomial expression, so that we can write down the expansion of without going through the lower powers.
Theorem: Binomial Theorem
For a positive integer ,
Here, the coefficients of the form , also written as , are called binomial coefficients.
Let us prove this theorem.
We can write
When we multiply through the parenthesis to write the expansion, we choose either or from each factor and multiply them together. For instance, if we choose 4 times and 6 times, this forms the term . Since these terms are added together, the number of different ways we can choose 4 times and 6 times becomes the coefficient of . Let us consider what these coefficients are.
First, note that there are factors in total, which means that we make a choice between and exactly different times. If we were to choose times, it means that we would need to choose times, which leads to the term . Then, the coefficient of this term is the number of different ways that we can choose exactly times.
For instance, we can consider how the term appears from the expansion of :
This just illustrates one instance of . We can choose from different factors to obtain the same term as follows:
We can see that as long as we choose 3 factors out of 8 total factors to assign to , we will end up with the term . This means that the number of different ways this term appears in the expansion is the same as the number of different ways to select 3 objects out of 8 distinct objects, where the order of the selected objects does not matter. We recall that this number is given by the combination rule, written as .
This argument can be generalized to show that the number of different times appears in the expansion of is the same as the number of different ways to choose of the factors from the total of factors, where the order of the selected factors does not matter. Hence, the coefficient of this term is given by . Summing these terms together with the coefficients, we obtain the statement of the binomial theorem. This completes the proof.
We can verify this theorem for the second and the third powers, which we have derived above. For the second power, we can write out the summation from the binomial theorem to write
We can compute
Also using and , we can write as expected. For the third power, we have
We can compute
This leads to as expected.
In our first example, we will apply the binomial theorem to expand the fourth power of a binomial expansion.
Example 1: Using the Binomial Theorem
Use the binomial theorem to find the expansion of .
Answer
In this example, we need to expand a binomial expression raised to the fourth power. We recall the binomial theorem, which states that, for a positive integer ,
Since our given expression is , we can apply the binomial theorem with , , and . Substituting these expressions into the binomial theorem, we have
Any power of 1 is equal to 1, so we can ignore in the summand. We can write out this sigma notation as the following sum:
Let us compute the binomial coefficients:
Also, using , we can substitute these values above to write the expansion of as
We can also apply the binomial theorem when a binomial expression is a difference rather than a sum. To apply the binomial theorem in this case, we need to write
In the next example, we will expand the third power of a binomial expression for a difference.
Example 2: Expanding a Binomial Using the Binomial Theorem
Expand .
Answer
In this example, we need to expand a binomial expression raised to the third power. We recall the binomial theorem, which states that, for a positive integer ,
Before we can apply the binomial theorem to our expression, we need to notice that our binomial expression is a difference rather than a sum. We can first rewrite our expression as
Then, we can apply the binomial theorem with , , and :
We can write out this sigma notation as a sum:
Let us compute the binomial coefficients:
Also, we know that
Substituting these values above, we obtain the expansion of as
In previous examples, we applied the binomial theorem to expand a power of a binomial. While this theorem provides a direct way to compute the expansion, it can be still burdensome to compute the binomial coefficients for higher powers. Fortunately, there is a symmetric structure of these coefficients, which we now recall.
Property: Symmetry of Binomial Coefficients
Given positive integers and satisfying , we have
This means that if we have computed the first half (including the middle term if there is an odd number of terms) of the binomial coefficients, the remaining coefficients can be deduced from what we already have computed.
In the next example, we will use this method to reduce the number of binomial coefficients we need to compute to expand a binomial expression raised to the fifth power.
Example 3: Expanding a Binomial Using the Binomial Theorem
Expand .
Answer
In this example, we need to expand a binomial expression raised to the fifth power. We recall the binomial theorem, which states that, for a positive integer ,
Before we can apply the binomial theorem to our expression, we need to notice that our binomial expression is a difference rather than a sum. We can first rewrite our expression as
Then, we can apply the binomial theorem with , , and :
We can write out this sigma notation as a sum:
Next, we need to compute the binomial coefficient . We can use the symmetric property of the binomial coefficients to reduce the number of coefficients we need to compute. Recall that given positive integers and satisfying , we have
This means that we only need to compute the first three coefficients:
Using the symmetric property, we have
Substituting these values and expanding individual powers, we can write the expansion as
Simplifying each fraction, we obtain
So far, we have focused on expanding a positive integer power of a binomial expression. We can also use the binomial theorem for the converse process, which is to factor a given polynomial expression, granted that the coefficients follow the exact pattern stated in the binomial theorem.
In the next example, we will factor a seventh-degree polynomial using the binomial theorem and use the factored form to find the root of the equation.
Example 4: Using Binomial Expansion to Find the Value of an Unknown
Find the value of that satisfies
Answer
In this example, we need to find the root of a seventh-degree polynomial, for which there is no general formula. However, the coefficients of the polynomial on the left-hand side of the equation appear to follow a familiar pattern, given by the binomial theorem. Recall the binomial theorem, which states that, for a positive integer ,
If we can factor the polynomial on the left-hand side of the equation to a power of a binomial expression, we will be able to find the root of the equation.
Let us observe a few patterns in the coefficients of this polynomial. We can see the power of 3 appears in the denominator, and we can write these as
The powers of 3 are the same as the corresponding powers of , so we can factor these powers together to write
By this point, we can see that the coefficients are beginning to resemble the expressions for binomial coefficients, which are . Since the highest power is the seventh power, we can assume that . It is now reasonable to conjecture that this expression is the expansion of , but we need to do a bit more work to know this for sure.
We can write the coefficients into this form by noting that , so the polynomial is written as
Hence, this is the same as
Using the binomial theorem, this polynomial can be factored as . This means that our equation can be written as
Since , we can take the seventh root of both sides of the equation to obtain
Rearranging this equation leads to the root .
In the previous example, we factored a seventh-degree polynomial to find the root of a given equation. This process can also lead to interesting properties involving the binomial coefficients, as we will see in the next example.
Example 5: Calculating the Sum of Binomial Coefficients
Find the value of .
Answer
In this example, we want to find the sum of the binomial coefficients of the form . While we can find this value by computing each binomial coefficient and summing, it is quicker to apply the binomial theorem for this example. Recall the binomial theorem, which states that, for a positive integer ,
To apply this theorem to the given expression, let us write the sigma notation as a sum when :
We can see that the right-hand side of this equation resembles the given expression, except that the given expression does not involve any powers of numbers. If we choose and , all powers of and will disappear. In this case, we can write this equation as
This tells us that the given expression is equal to . Hence,
In the previous example, we found the value of the sum of binomial coefficients by applying the binomial theorem. We can state this fact for a general power , which corresponds to writing out the binomial theorem for general with . This leads to the following property.
Property: Sum of Binomial Coefficients
Given a positive integer , we have
We now turn our attention to how to use the binomial theorem to approximate a positive integer power of a real number. When we want to compute a real number raised to a positive integer power, we can use the binomial theorem to approximate this value. To do this, we need to first write the given number as its integer part plus its decimal part and write the power as , where is an integer and . For instance, we can write
We can note that is not always positive, and the integer is not always less than the given number. Instead, it is more beneficial to choose the nearest integer to the given number and then assign accordingly. Then, we can write out the expansion of this power using the binomial theorem:
Since , we know that the powers of will become smaller as we progress through the expansion. Also, since is an integer, the powers of will decrease in size since we are taking smaller powers of as we progress through the expansion. While the binomial coefficients may vary in size, these behaviors of different powers of and result in the general trend that the terms will decrease in size as we progress through this binomial expansion. In other words, the next term in the binomial series will almost always be smaller than the previous term.
Hence, in order to gain a specified accuracy, we do not have to compute the entire expansion. For instance, to gain the accuracy of three decimal places, it is often sufficient to stop computing when the size of the next term is smaller than 0.0001.
In the final example, we will use this method to approximate a power of a number close to 1.
Example 6: Approximating the Value of a Numerical Expression Using the Binomial Theorem
Using the binomial theorem, approximate to three decimal places the value of .
Answer
In this example, we want to use the binomial theorem to approximate the power of a number that is close to 1. We recall that the binomial theorem tells us that
To approximate , we can begin by writing
We can expand this power using the binomial theorem,
Note that the higher powers of 0.05 such as are small compared to the first two terms of the sum on the right-hand side of this equation. Hence, the terms in this expansion will likely decrease in size as we progress through the expansion. Since we want to approximate this number up to three decimal places, we will stop computing when a term is below 0.0001. We compute the terms
We can omit the remaining terms since the previous term is already below the threshold of 0.0001. This leads to
Rounding to the nearest thousandth, we have approximated by 1.340.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- For a positive integer , Here, the coefficients of the form , also written as , are called binomial coefficients.
- We can expand the positive integer power of a binomial expression using the binomial theorem. We can also factor certain polynomials using the binomial theorem.
- Due to the symmetric property of binomial coefficients , we only need to compute the first half (plus the middle term if there is an odd number of terms) of the binomial coefficients.
- Using the binomial theorem, we can approximate a positive integer power of a number by writing it as , where is the closest integer to the number and satisfies .
