The Binomial Theorem | Moosa Academy

Expanding by multiplying out five brackets is slow and error-prone. The binomial theorem writes the answer down directly. It has only two moving parts — the coefficients, which come from Pascal's Triangle, and the exponents, which follow a fixed pattern in every single case.

Concept When the theorem applies

The binomial theorem expands any expression of the form , provided there are exactly two terms inside the bracket and is a positive whole number. The small cases are worth recognising on sight:

Part 1 The coefficients

The numbers in front of each term are read straight off Pascal's Triangle — or equivalently calculated as combinations. Pick the row matching your value of :

For

, take the row

. Those five numbers are the coefficients, in order.

Part 2 How the exponents move

The exponents follow the same pattern in every expansion:

The exponent on

starts at

and decreases:

down to 0.

The exponent on

starts at 0 and increases up to

In every term, the two exponents add up to .

Take . Ignoring coefficients, the terms are:

The power of falls, the power of rises, and each pair sums to 5. That last fact is the quickest way to check your work.

Concept How many terms

The expansion of always has terms.

gives 3 terms.

gives 4 terms.

gives 5 terms.

gives 6 terms.

This is also how you pick the right row of Pascal's Triangle: choose the row containing exactly numbers.

Concept The general formula

is the coefficient — the entry from Pascal's Triangle, or the combination "n choose k".

carries the falling exponent.

carries the rising one.

Their exponents sum to

, exactly as the pattern requires.

The formula is simply the two patterns written in one line. Reading it that way makes it far easier to remember than treating it as a symbol to memorise.

Example Expanding (a + b)⁵

, so there will be 6 terms.

Take row 5 of Pascal's Triangle:

Write the powers of

falling from 5 and the powers of

rising from 0.

Attach each coefficient to its term in order.

⟹ a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Example When the second term is a number

Expand .

Here

and

, with

Row 4 of the triangle gives

The powers of 3 must be worked out:

Multiply each coefficient by the matching power:

⟹ x⁴ + 12x³ + 54x² + 108x + 81

Whenever is a number rather than a letter, remember to raise it to the power as well — leaving it as a plain 3 is the most common error here.

Note Mistakes to avoid

Using the wrong row of Pascal's Triangle — the row must hold

numbers.

Letting both exponents rise, or both fall; one always does the opposite of the other.

Ending up with a term whose exponents do not add to

Forgetting to raise a numerical second term to its power.

Writing

terms instead of

Applying the theorem to a bracket with three terms — it is for binomials only.

Summary

The binomial theorem expands (a + b)ⁿ for a positive whole number n.
The coefficients come from the matching row of Pascal's Triangle, or from combinations.
The exponent on a falls from n to 0 while the exponent on b rises from 0 to n.
In every term the two exponents add up to n — a quick way to check your work.
The expansion always has n + 1 terms, which is also how you choose the right row.