Binomial Theorem General Term: Find Any Coefficient

Suppose a question asks for the coefficient of x^4 in (2x - 3)^7. You could expand all eight terms, but that would mean doing seven pieces of work you were never asked to do.

The binomial theorem gives you a way to go straight to the term you need. The trick is to find its address first. Once you know which value of r produces x^4, the arithmetic is quite ordinary.

In this guide, we will use the general term to find particular coefficients, constant terms, and even powers that do not appear at all. You will also see why the fourth term and the term containing x^4 are different requests.

The general term formula, explained gently

For a non-negative integer n, the binomial expansion of (a + b)^n has the general term:

T_(r+1) = C(n, r) × a^(n-r) × b^r, where r = 0, 1, 2, …, n.

Here, T_(r+1) means the term in position r + 1. We use a caret for powers, so x^3 means x cubed. The notation C(n, r), read as “n choose r”, means:

C(n, r) = n! / [r!(n - r)!]

The exclamation mark means factorial: 5! = 5 × 4 × 3 × 2 × 1. Also, 0! = 1.

Why does this formula work? Imagine multiplying n copies of (a + b). To obtain a^(n-r)b^r, you must choose b from exactly r of those brackets and a from the others. C(n, r) counts how many choices produce that same product. The University of Lethbridge’s combinatorics text develops this counting explanation.

The indexing starts at zero because the first term contains no factors of b:

Term position	Value of r	Term
First	0	a^n
Second	1	n × a^(n-1) × b
Third	2	C(n, 2) × a^(n-2) × b^2
Last, position n + 1	n	b^n

If the question asks for the sixth term, use r = 5. If it asks for a particular power of x, you must first work out which r produces that power.

Term, coefficient, and binomial coefficient are different

Consider the term -75x^4.

• The complete term is -75x^4.
• The coefficient of x^4 is -75.
• The exponent of x is 4.

A binomial coefficient such as C(7, 3) is only one factor in the final coefficient. Constants from a and b, including their signs, must also be included.

For instance, in (2x + 3)^7, the number 2 belongs to the first entry and 3 belongs to the second entry. Replacing the entries by x and 1 changes the question.

It helps to think of the general term as two separate parts:

numerical factor × power of x

The power tells you whether you have reached the right term. The numerical factor gives you the coefficient once you get there.

A reliable five-step method

1. Identify the complete first and second entries inside the brackets.
2. Substitute them into T_(r+1), keeping brackets around each entry.
3. Collect all powers of x into one power.
4. Set that exponent equal to the required exponent and solve for r.
5. Check that r is an integer from 0 to n, then calculate the numerical factor.

For a constant term, the required exponent is zero. For the coefficient of 1/x^3, it is -3.

If no permitted integer r works, the requested power is absent and its coefficient is zero. Do not round r to the nearest whole number. A different r means a different term.

Example 1: Find the coefficient of x^4 in (2x - 3)^7

The two entries are a = 2x and b = -3. Keep that minus sign attached to 3.

The general term is:

T_(r+1) = C(7, r) × (2x)^(7-r) × (-3)^r

Separate numbers from the variable:

T_(r+1) = C(7, r) × 2^(7-r) × (-3)^r × x^(7-r)

We want x^4, so:

7 - r = 4, giving r = 3.

This is a permitted value. The required term is therefore T_4, and its coefficient is:

C(7, 3) × 2^4 × (-3)^3

= 35 × 16 × (-27)

= -15,120

The complete term is -15,120x^4, but the answer to the question is just -15,120.

Notice that the sign is negative because r is odd. Although the outer power is 7, the sign of this particular term is controlled by the power placed on -3.

Example 2: Find the fifth term in (3x + 2)^6

This question gives a position, so there is no exponent equation to solve.

For the fifth term, r = 4:

T_5 = C(6, 4) × (3x)^2 × 2^4

= 15 × 9x^2 × 16

= 2,160x^2

Compare that with a different question: find the coefficient of x^5 in the same expansion.

Now the exponent equation is 6 - r = 5, so r = 1. The coefficient becomes:

C(6, 1) × 3^5 × 2 = 2,916

The fifth term contains x^2, while the x^5 term is the second term. Reading the request carefully saves more time than any shortcut.

Example 3: A constant term with reciprocal powers

Find the term independent of x in (2x^2 + 1/x)^6, where x is nonzero.

First write 1/x as x^(-1), or keep the fraction and apply the power to its denominator:

T_(r+1) = C(6, r) × (2x^2)^(6-r) × (1/x)^r

= C(6, r) × 2^(6-r) × x^(12-2r) × x^(-r)

= C(6, r) × 2^(6-r) × x^(12-3r)

A term independent of x has exponent zero:

12 - 3r = 0, so r = 4.

The constant term is:

C(6, 4) × 2^2 = 15 × 4 = 60

It is the fifth term in the standard expansion order.

A common mistake is to write the exponent as 12 - 2r. That accounts for the first entry but forgets the x^(-r) contributed by the second entry. Both entries affect the final exponent.

Example 4: When the requested power is missing

Find the coefficient of x^5 in (x^2 + 3)^5.

The general term is:

T_(r+1) = C(5, r) × 3^r × x^(10-2r)

For x^5, we would need:

10 - 2r = 5, giving r = 5/2.

But r must be an integer. Therefore, the coefficient of x^5 is 0.

There is also a quick visual check. The possible powers are 10, 8, 6, 4, 2, and 0. Each step replaces one x^2 with a constant, so the exponent decreases by 2. No odd power can appear.

An integer answer for r is not enough on its own, either. Asking for x^12 in this expansion gives r = -1. That is an integer, but it falls outside 0 through 5, so its coefficient is also zero.

Example 5: Find a negative-power coefficient

Find the coefficient of x^(-3) in (x^2 - 2/x)^6, where x is nonzero.

The general term becomes:

T_(r+1) = C(6, r) × (x^2)^(6-r) × (-2/x)^r

= C(6, r) × (-2)^r × x^(12-3r)

Match the exponent to -3:

12 - 3r = -3, so r = 5.

The coefficient is:

C(6, 5) × (-2)^5 = 6 × (-32) = -192

The term is -192x^(-3), which can also be written as -192/x^3.

Negative powers of x are perfectly possible here. What must stay non-negative is the integer index r, along with the integer outer power n in this finite expansion.

A compact formula for mixed powers

Once the method feels comfortable, you can organise it in one line.

For (Ax^p + Bx^q)^n, the general term is:

T_(r+1) = C(n, r) × A^(n-r) × B^r × x^[pn + (q-p)r]

For the coefficient of x^m, solve:

pn + (q - p)r = m

When p and q differ:

r = (m - pn)/(q - p)

Here A and B are constants, with any minus sign included in B. For the integer powers used in this guide, negative exponents require x to be nonzero.

For the common form (Ax^p + B/x^q)^n, the exponent is instead written as pn - (p + q)r, with q representing the positive denominator power.

You do not have to memorise both versions. Writing the powers contributed by each entry is often safer.

There is one useful exception to the single-term shortcut. If p = q, all the formal terms have the same power of x. Combine the entries first:

(Ax^p + Bx^p)^n = (A + B)^n × x^(pn)

For example, (2x + 3x)^4 = 625x^4. Selecting just one formal term would miss the other contributions to the coefficient.

Example 6: Two variables must agree on the same r

Find the coefficient of x^6y^3 in (2x^2 - y)^6.

The general term is:

T_(r+1) = C(6, r) × 2^(6-r) × (-1)^r × x^(12-2r) × y^r

The y exponent requires r = 3. Check that this also gives the required x exponent:

12 - 2(3) = 6.

Both requirements agree. The coefficient is:

C(6, 3) × 2^3 × (-1)^3 = -160

Now suppose the question asked for x^6y^2 instead. The x exponent would require r = 3, while the y exponent would require r = 2. No single term can satisfy both, so that coefficient is zero.

With two variables, checking only one exponent can lead you to a convincing but incorrect answer.

Example 7: A coefficient containing a parameter

If the coefficient of x^2 in (1 + kx)^5 is 90 and k is positive, find k.

Here the power of x in the general term is r, so use r = 2:

Coefficient of x^2 = C(5, 2) × k^2 = 10k^2.

Therefore:

10k^2 = 90

k^2 = 9

Since k is positive, k = 3.

Without the positivity condition, both k = 3 and k = -3 would work. Matching a coefficient can give more than one parameter value, so read every condition in the question.

How to present a clear written solution

A good solution makes the choice of r visible. You usually need four short stages:

Stage	What to show
General term	Substitute the complete entries with their signs
Power matching	Write the exponent equation
Validity check	Confirm r is an integer between 0 and n
Final answer	Give the coefficient or complete term requested

Avoid expanding every term merely to demonstrate effort. The general term already explains why your selected term is the right one.

For larger combinations, cancel factorials before multiplying. For example:

C(12, 3) = (12 × 11 × 10)/(3 × 2 × 1) = 220.

There is no need to evaluate 12! and 9! separately. The identity C(n, r) = C(n, n-r) also lets you use the smaller calculation. For instance, C(12, 10) = C(12, 2) = 66.

If your question asks for a central position rather than a particular power, use the companion guide to middle terms in binomial expansion.

Common mistakes and quick repairs

Mistake	Why it fails	Repair
Using r = 5 for the fifth term	The first term uses r = 0	Use r = 4
Reporting only C(n, r)	Numerical factors inside the brackets also contribute	Include both entries’ numerical powers
Dropping a minus sign	The sign is raised to r along with the second entry	Write the negative entry in brackets
Forgetting a denominator’s power	Reciprocal powers affect the final exponent	Rewrite 1/x^q as x^(-q)
Rounding a fractional r	Each integer r identifies a different term	Conclude that the requested power is absent
Returning a term for a coefficient question	The question asks for the multiplier only	Remove the requested variable power from the answer

One final check is to list just the possible exponents. You do not need their coefficients. This small list often catches an impossible target or an incorrect sign in the exponent equation.

Try these before checking the answers

1. Find the coefficient of x^3 in (x + 2)^5.
2. Find the fourth term in (2x - 1)^5.
3. Find the constant term in (x^2 + 2/x)^6, for nonzero x.
4. Find the coefficient of x^7 in (x^2 + 1)^6.
5. Find the coefficient of x^(-2) in (x - 1/x)^8, for nonzero x.
6. Find the coefficient of x^4y^2 in (3x - 2y)^6.
7. The coefficient of x^3 in (1 + kx)^4 is 108. Find the real value of k.
8. Find the coefficient of x^2 in (2x + 3x)^2.

Answers with the key working

1. 40. Match 5 - r = 3, so r = 2. The coefficient is C(5, 2) × 2^2 = 40.

2. -40x^2. The fourth term uses r = 3. Thus T_4 = C(5, 3) × (2x)^2 × (-1)^3 = -40x^2.

3. 240. The exponent is 12 - 3r. Setting it to zero gives r = 4, so the constant is C(6, 4) × 2^4 = 240.

4. 0. The equation 12 - 2r = 7 gives r = 5/2. No permitted integer r exists.

5. -56. Solve 8 - 2r = -2 to get r = 5. The coefficient is C(8, 5) × (-1)^5 = -56.

6. 4,860. The y exponent gives r = 2, and the x exponent is then 4. Calculate C(6, 2) × 3^4 × (-2)^2 = 4,860.

7. 3. The coefficient is C(4, 3)k^3 = 4k^3. Hence k^3 = 27 and k = 3.

8. 25. Combine the entries first: (2x + 3x)^2 = (5x)^2 = 25x^2. All formal expansion terms contribute to the same power.

Further reading

The Australian Mathematical Sciences Institute’s lesson on applying the binomial theorem offers further examples of general terms and constant terms. For the counting argument behind the formula, revisit the University of Lethbridge’s binomial theorem section.

Frequently asked questions

What is the general term of a binomial expansion?

For (a + b)^n with non-negative integer n, it is T_(r+1) = C(n, r) × a^(n-r) × b^r, where r runs through the integers from 0 to n. Substituting a permitted r gives one term in the standard expansion order.

Why is it T_(r+1) rather than T_r?

The first term contains b^0, so it uses r = 0. Its position is 1. The term position is therefore always one more than r in this notation.

How do I find a coefficient without expanding everything?

Write the general term and collect the variable powers. Set the resulting exponent equal to the required exponent, solve for r, check its validity, and calculate the numerical factor.

Is the coefficient of x^4 always found using r = 4?

No. That works when the general term contains x^r, as in (1 + ax)^n. In (ax + b)^n, the power is x^(n-r). With powers or reciprocals in both entries, the exponent may be more complicated.

What does term independent of x mean?

It means a term containing x^0, which is 1 for nonzero x. Set the total exponent of x equal to zero. In a polynomial, this is the usual constant term, and the simplified polynomial is also defined at x = 0.

What if r is a fraction or is outside 0 through n?

The requested power does not occur, so its coefficient is zero. Do not round r or substitute an invalid value into the finite binomial coefficient formula.

Can the coefficient be negative?

Yes. A negative entry inside the brackets may produce negative coefficients. Carry its sign through the relevant power before calculating the final number.

Is C(n, r) the complete coefficient?

Only in special forms such as (1 + x)^n. In general, multiply it by the numerical factors contributed by both entries. Those factors can change the size and sign substantially.

Can I reverse the order of the two entries?

Yes, the resulting expression is the same, but the standard term order reverses. A coefficient of a specified power stays the same. For a question asking for the kth term, preserve the stated order unless another order is explicitly requested.

What is the fastest way to check my answer?

Substitute your r back into the exponent expression, check that it lies from 0 to n, and verify the sign separately. Then check whether the question wants the coefficient alone or the entire term. These small checks catch most common mistakes.