In this explainer, we will learn how to expand any binomial expression of the form .
When we learn elementary algebra, we learn how to factor and expand binomials raised to the second power. Using similar techniques we are theoretically able to expand binomials raised to arbitrarily large powers. However, we soon find that this becomes extremely laborious even for relatively small powers. Fortunately, we are not confined to this approach, and there is an important theorem, known as the binomial theorem, which gives us a general form for expanding binomials.
Binomial Theorem
For an integer , where
Sometimes the following notations are used in place of : , , , , and .
In this explainer, we will learn how to apply this theorem to expand binomials and find the coefficients of specific terms within an expansion. We will begin by considering a couple of examples where we apply the binomial theorem to fully expand binomials raised to a positive integer power.
Example 1: Applying the Binomial Theorem
Use the binomial theorem to find the expansion of .
Answer
Recall the binomial theorem which states that, for an integer ,
Setting , we have
Using a calculator, or Pascalโs triangle, we can evaluate the combinations; we can rewrite this as
As we can see, applying the binomial theorem can seriously simplify expanding binomials. However, we need to be a little careful when we have constants multiplying each term, as the next example will demonstrate.
Example 2: Using the Binomial Theorem
Expand .
Answer
Before we apply the binomial theorem. It is useful to check for common factors in our terms. In this case, there is a common factor of 2. Hence, before we get into the details of the binomial expansion, we should factor out the 2 as follows:
We can now apply the binomial theorem to . Recall that the binomial theorem states that, for an integer ,
Setting , , and , we have
We can either use Pascalโs triangle or a calculator to evaluate the combinations . We can then rewrite this as
Since we can use our expansion of to rewrite this as
Sometimes we are not expected to give the whole expansion: we might be asked for the first terms, or for a specific term. When we are asked for a specific term, it is often useful to use the formula for the general term.
General Term of the Binomial Expansion
In the expansion of , the general term is given by
The general term is variously notated or . The important thing to note when referring to terms by order is that the first term is the term for which . Hence,
In the next example, we will consider a question which is best solved using the general term.
Example 3: Finding the Coefficient of the General Term
Find the coefficient of the fourth term in the expansion of .
Answer
When presented with a question like this, it would be perfectly legitimate to write the full expansion and then take the coefficient from the appropriate term. However, appealing to the formula for the general termsimplifies our calculation. This is the method we will demonstrate here. Recall that the formula for the general term for the expansion is
However, we need to remember that the first term starts from . Hence, the fourth term will be given by and not, as we might naรฏvely think, by . Hence, its coefficient will be given by .
Example 4: Finding the Coefficient of the General Term
Find the coefficient of in the expansion of .
Answer
The easiest way to approach this problem is to consider the general term for the expansion :
We begin by setting , and which gives a general term of
We can simplify this to get
Since we are looking for the coefficient of the term , we can solve
for , which gives . Therefore, the coefficient of this term will be given by
By evaluating this in a calculator, we have that the coefficient of the term in the expansion of is 2,880.
Having got a basic grounding in working with binomial expansions, we can now turn our attention to problems which involve finding unknowns.
Example 5: Finding an Unknown Constant
Answer the following questions for the expansion of .
- Given that the coefficient of is 60, and is positive, find .
- Hence, using your value of , work out the coefficient of in the expansion.
Answer
Part 1
Recall that the general term of the expansion is
Setting , , and , we have that the general term of is
We are interested in the coefficient of when . Hence, we have
Simplifying, we have
Hence,
Since we know is positive, we can take the positive square root to get
Part 2
Now we can use this value of to find the coefficient of in the expansion of . We know that the general term in the expansion is given by
Notice that when . Hence, the coefficient of will be given by
Evaluating this expression, we get that its coefficient in the expansion of is .
Example 6: Finding an Unknown Power
Answer the following questions for the expansion of .
- Given that the coefficient of is 3,840, find .
- Hence, work out the value of the coefficient of .
Answer
Part 1
This problem is easiest to solve using the formula for the general term of the binomial expansion. Recall that for this is given by
Setting and , we have that the general term of is
When , we have an expression for the coefficient of the term. Hence,
Using the definition of , we can rewrite this as
To solve this equation for , we consider the prime factors of 3,840. By successively dividing by 2, we find that . This is not in the form we would like, since we are looking for two consecutive integers multiplied together multiplied by a power of 2. By moving one of the powers of two to multiply the three, we can rewrite this as . Comparing this with , we can see that is a solution to this equation.
Part 2
Having found the value of , we can find the coefficient of using the general term by setting . This gives the coefficient as
Evaluating this, we have that the coefficient of in the expansion of is 12,288.
The binomial theorem helps us expand powers of binomial. However, it can also help us expand trinomials; to do this we use the technique of using parentheses to express the trinomial as a binomial which we can expand. The next example will demonstrate this technique.
Example 7: Dealing with Parentheses with Three Terms
Given that the first three terms in the expansion of are 1, , and , determine the values of and .
Answer
In this question, we have been given a trinomial to expand. Therefore, we cannot simply apply the binomial theorem. Instead, we write , which gives us a binomial which we can expand using the binomial theorem. We are only interested in the first three terms of the fully expanded expression. Therefore, we begin by using the binomial theorem to write out the first three terms of the expansion of :
Notice that the terms after these three all contain higher powers of . Therefore, they will not contribute to the constant term and the first two powers of . Expanding the parentheses in the second and third term, we have
Since we are only interested in the constant term and the coefficients of and , we can ignore the terms with higher powers of and gather our like terms as follows:
Comparing the terms, we have . Therefore, . We can now consider the term, using the value of ; we have
Hence,
Subtracting 66 from both sides, we have
Then, by dividing by 12, we get . Hence, and .
Finally, we consider a couple of questions where we look at using the binomial theorem in reverse.
Example 8: Binomial Theorem in Reverse
Find all possible real values of satisfying
Answer
Solving a problem like this relies on us being able to recognize an expression as a binomial expansion. The first thing we notice is that consecutive terms have decreasing powers of and increasing powers of 2. However, we need to be careful that the terms are alternating in sign. Hence, one of the terms in our binomial expansion will be negative. In fact, it can be useful to factor into one of the terms or another. We notice that the negative terms correspond to the odd powers of 2; therefore, we can rewrite the expression on the left-hand side as follows:
Furthermore, we notice that the coefficients are precisely the binomial coefficients for the expansion of a binomial to the ninth power. If this were not the case, we would see whether we could find a common factor to transform them into binomial coefficients. Using the notation for binomial coefficients, we can rewrite this as
Therefore, we can factor the expression on the left-hand side to
Taking the ninth root, we have
Hence,
Example 9: Reverse Binomial Theorem
Find the coefficient of in the expansion of .
Answer
We could attempt to solve this problem by expanding each of the sets of parentheses individually. However, this would be laborious and prone to mistake, whereas careful inspection of the given expression highlights a more optimal solution. The expression is a binomial expansion in two terms and . Hence, we can rewrite the expression as
We can now use the general term to find the coefficient of . Recall that, in the expansion of , the general term is given by
Setting , , , and , we have that the term is given by
Evaluating this numerical expression, we find that the coefficient of is given by 70,000.
Key Points
- The binomial theorem states that, for an integer , where
- Using the binomial theorem enables us to expand binomials raised to an arbitrarily large exponent.
- In the expansion of , the general term is given by The general term is variously notated or .
- Familiarity with binomial expansions can help us recognize expressions which are binomials in expanded form.