Lesson Explainer: The Binomial Theorem | Nagwa Lesson Explainer: The Binomial Theorem | Nagwa

Lesson Explainer: The Binomial Theorem Mathematics • Third Year of Secondary School

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In this explainer, we will learn how to expand any binomial expression of the form (𝑎+𝑏) using combinations.

We can expand a binomial expression raised to the second power as (𝑎+𝑏)=(𝑎+𝑏)(𝑎+𝑏)=𝑎+𝑎𝑏+𝑎𝑏+𝑏=𝑎+2𝑎𝑏+𝑏.

We can use this expression to expand a binomial expression raised to the third power as (𝑎+𝑏)=(𝑎+𝑏)(𝑎+𝑏)=𝑎+2𝑎𝑏+𝑏(𝑎+𝑏)=𝑎+2𝑎𝑏+𝑎𝑏+𝑎𝑏+2𝑎𝑏+𝑏=𝑎+3𝑎𝑏+3𝑎𝑏+𝑏.

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 (𝑎+𝑏) without going through the lower powers.

Theorem: Binomial Theorem

For a positive integer 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

Here, the coefficients of the form 𝐶, also written as 𝑛𝑟, are called binomial coefficients.

Let us prove this theorem.

We can write (𝑎+𝑏)=(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏).Repeatedtimes

When we multiply through the parenthesis to write the expansion, we choose either 𝑎 or 𝑏 from each factor and multiply them together. For instance, if we choose 𝑎 4 times and 𝑏 6 times, this forms the term 𝑎𝑏. Since these terms are added together, the number of different ways we can choose 𝑎 4 times and 𝑏 6 times becomes the coefficient of 𝑎𝑏. Let us consider what these coefficients are.

First, note that there are 𝑛 factors in total, which means that we make a choice between 𝑎 and 𝑏 exactly 𝑛 different times. If we were to choose 𝑏𝑟 times, it means that we would need to choose 𝑎𝑛𝑟 times, which leads to the term 𝑎𝑏. Then, the coefficient of this term is the number of different ways that we can choose 𝑏 exactly 𝑟 times.

For instance, we can consider how the term 𝑎𝑏 appears from the expansion of (𝑎+𝑏): (𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏).chosentimeschosentimes

This just illustrates one instance of 𝑎𝑏. We can choose 𝑏 from different factors to obtain the same term as follows: (𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏)(𝑎+𝑏).chosentimeschosentimes

We can see that as long as we choose 3 factors out of 8 total factors to assign to 𝑏, we will end up with the term 𝑎𝑏. 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 𝐶.

This argument can be generalized to show that the number of different times 𝑎𝑏 appears in the expansion of (𝑎+𝑏) is the same as the number of different ways to choose 𝑟 of the factors from the total of 𝑛 factors, where the order of the selected factors does not matter. Hence, the coefficient of this term is given by 𝐶. 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 (𝑎+𝑏)=𝐶𝑎𝑏+𝐶𝑎𝑏+𝐶𝑎𝑏.

We can compute 𝐶=2200=1,𝐶=2211=2,𝐶=2222=1.

Also using 𝑏=1 and 𝑎=1, we can write (𝑎+𝑏)=𝑎+2𝑎𝑏+𝑏, as expected. For the third power, we have (𝑎+𝑏)=𝐶𝑎𝑏+𝐶𝑎𝑏+𝐶𝑎𝑏+𝐶𝑎𝑏.

We can compute 𝐶=3300=1,𝐶=3311=3,𝐶=3322=3,𝐶=3333=1.

This leads to (𝑎+𝑏)=𝑎+3𝑎𝑏+3𝑎𝑏+𝑏, 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 (1+𝑥).

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 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

Since our given expression is (1+𝑥), we can apply the binomial theorem with 𝑎=1, 𝑏=𝑥, and 𝑛=4. Substituting these expressions into the binomial theorem, we have (1+𝑥)=𝐶1𝑥.

Any power of 1 is equal to 1, so we can ignore 1 in the summand. We can write out this sigma notation as the following sum: 𝐶𝑥+𝐶𝑥+𝐶𝑥+𝐶𝑥+𝐶𝑥.

Let us compute the binomial coefficients: 𝐶=4400=1,𝐶=4411=4,𝐶=4422=6,𝐶=4433=4,𝐶=4444=1.

Also, using 𝑥=1, we can substitute these values above to write the expansion of (1+𝑥) as 1+4𝑥+6𝑥+4𝑥+𝑥.

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 (𝑎𝑏)=(𝑎+(𝑏))=𝐶𝑎(𝑏).

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 𝑥2.

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 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

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 𝑥2=𝑥+2.

Then, we can apply the binomial theorem with 𝑎=𝑥, 𝑏=2, and 𝑛=3: 𝑥+2=𝐶𝑥2.

We can write out this sigma notation as a sum: 𝐶𝑥2+𝐶𝑥2+𝐶𝑥2+𝐶𝑥2.

Let us compute the binomial coefficients: 𝐶=3300=1,𝐶=3311=3,𝐶=3322=3,𝐶=3333=1.

Also, we know that 𝑥=1,2=1,2=2,2=2,2=22.

Substituting these values above, we obtain the expansion of 𝑥2 as 𝑥32𝑥+6𝑥22.

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 𝐶 for higher powers. Fortunately, there is a symmetric structure of these coefficients, which we now recall.

Property: Symmetry of Binomial Coefficients

Given positive integers 𝑛 and 𝑟 satisfying 𝑛𝑟, we have 𝐶=𝐶.

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 𝑥41𝑥.

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 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

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 𝑥41𝑥=𝑥4+1𝑥.

Then, we can apply the binomial theorem with 𝑎=𝑥4, 𝑏=1𝑥, and 𝑛=5: 𝑥41𝑥=𝐶𝑥41𝑥.

We can write out this sigma notation as a sum: 𝐶𝑥41𝑥+𝐶𝑥41𝑥+𝐶𝑥41𝑥+𝐶𝑥41𝑥+𝐶𝑥41𝑥+𝐶𝑥41𝑥.

Next, we need to compute the binomial coefficient 𝐶. 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 𝑛 and 𝑟 satisfying 𝑛𝑟, we have 𝐶=𝐶.

This means that we only need to compute the first three coefficients: 𝐶=5500=1,𝐶=5511=5,𝐶=5522=10.

Using the symmetric property, we have 𝐶=𝐶=10,𝐶=𝐶=5,𝐶=𝐶=1.

Substituting these values and expanding individual powers, we can write the expansion as 𝑥45𝑥41𝑥+10𝑥41𝑥10𝑥41𝑥+5𝑥41𝑥1𝑥.

Simplifying each fraction, we obtain 𝑥10245𝑥256+5𝑥3258𝑥+54𝑥1𝑥.

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 𝑥 that satisfies 1+73𝑥+7×692𝑥+7×6×5273𝑥++12187𝑥=2187.

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 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

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 1+73𝑥+7×632𝑥+7×6×533𝑥++13𝑥.

The powers of 3 are the same as the corresponding powers of 𝑥, so we can factor these powers together to write 1+71𝑥3+7×62𝑥3+7×6×53𝑥3++𝑥3.

By this point, we can see that the coefficients are beginning to resemble the expressions for binomial coefficients, which are 𝐶=𝑛𝑛𝑟𝑟. Since the highest power is the seventh power, we can assume that 𝑛=7. It is now reasonable to conjecture that this expression is the expansion of 1+𝑥3, 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 𝐶=𝐶=1, so the polynomial is written as 𝐶1𝑥3+7611𝑥3+7521𝑥3+7431𝑥3++𝐶1𝑥3.

Hence, this is the same as 𝐶1𝑥3+𝐶1𝑥3+𝐶1𝑥3+𝐶1𝑥3++𝐶1𝑥3.

Using the binomial theorem, this polynomial can be factored as 1+𝑥3. This means that our equation can be written as 1+𝑥3=2187.

Since 2187=3, we can take the seventh root of both sides of the equation to obtain 1+𝑥3=3.

Rearranging this equation leads to the root 𝑥=6.

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 𝐶+𝐶+𝐶++𝐶.

Answer

In this example, we want to find the sum of the binomial coefficients of the form 𝐶. 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 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where

To apply this theorem to the given expression, let us write the sigma notation as a sum when 𝑛=5: (𝑎+𝑏)=𝐶𝑎𝑏+𝐶𝑎𝑏+𝐶𝑎𝑏++𝐶𝑎𝑏.

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 𝑎=1 and 𝑏=1, all powers of 𝑎 and 𝑏 will disappear. In this case, we can write this equation as (1+1)=𝐶+𝐶+𝐶++𝐶.

This tells us that the given expression is equal to (1+1)=2=32. Hence, 𝐶+𝐶+𝐶++𝐶=32.

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 𝑛, which corresponds to writing out the binomial theorem for general 𝑛 with 𝑎=𝑏=1. This leads to the following property.

Property: Sum of Binomial Coefficients

Given a positive integer 𝑛, we have 𝐶+𝐶++𝐶=2.

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 (𝑎+𝑥), where 𝑎 is an integer and |𝑥|<1. For instance, we can write 5.21=(5+0.21),1.89=(20.11).

We can note that 𝑥 is not always positive, and the integer 𝑎 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 𝑥 accordingly. Then, we can write out the expansion of this power using the binomial theorem: (𝑎+𝑥)=𝐶𝑎+𝐶𝑎𝑥++𝐶𝑥.

Since |𝑥|<1, we know that the powers of 𝑥 will become smaller as we progress through the expansion. Also, since 𝑎 is an integer, the powers of 𝑎 will decrease in size since we are taking smaller powers of 𝑎 as we progress through the expansion. While the binomial coefficients 𝐶 may vary in size, these behaviors of different powers of 𝑎 and 𝑥 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 (1.05).

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 (1+𝑥)=𝐶+𝐶𝑥+𝐶𝑥++𝐶𝑥.

To approximate (1.05), we can begin by writing (1.05)=(1+0.05).

We can expand this power using the binomial theorem, (1+0.05)=𝐶+𝐶(0.05)+𝐶(0.05)++𝐶(0.05).

Note that the higher powers of 0.05 such as (0.05),,(0.05) 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 𝐶=1,𝐶(0.05)=6×0.05=0.3,𝐶(0.05)=15×0.05=0.0375,𝐶(0.05)=20×0.05=0.0025,𝐶(0.05)=15×0.05=0.00009.

We can omit the remaining terms since the previous term is already below the threshold of 0.0001. This leads to (1+0.05)1+0.3+0.0375+0.0025+0.00009=1.34009.

Rounding to the nearest thousandth, we have approximated (1.05) by 1.340.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • For a positive integer 𝑛, (𝑎+𝑏)=𝐶𝑎𝑏,𝐶=𝑛𝑛𝑟𝑟.where Here, the coefficients of the form 𝐶, also written as 𝑛𝑟, 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 𝐶=𝐶, 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 (𝑎+𝑥), where 𝑎 is the closest integer to the number and 𝑥 satisfies |𝑥|<1.

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