Further Mathematics · Lesson 8

The binomial theorem

WAECNECO

What the syllabus asks

WAEC: Binomial theorem (expansion of (a + b)ⁿ for positive integral n, Pascal's triangle, ⁿCᵣ, general term, approximations). Supports UTME Mathematics.

NECO: Binomial theorem (expansion of (a + b)ⁿ for positive integral n, Pascal's triangle, ⁿCᵣ, general term, approximations). Supports UTME Mathematics.

What you will learn

Expand small powers of a binomial by repeated multiplication and spot the patterns.
Build and use Pascal's triangle.
Evaluate factorials and ⁿCᵣ.
State and apply the binomial theorem for positive integer n.
Use the general term to find a particular term, a coefficient, or the term independent of x.
Use binomial expansions for approximations, and solve problems with unknown n or k.

Key points

(a + b)ⁿ has n + 1 terms; the powers of a fall and the powers of b rise, adding to n in every term.
Pascal's triangle: each entry is the sum of the two above; row n gives the coefficients.
ⁿCᵣ = n!/[r!(n − r)!]; ⁿCᵣ = ⁿCₙ₋ᵣ.
General term: Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ (the kth term has r = k − 1).
Constant term: set the total power of x to 0.
For small x: (1 + x)ⁿ ≈ 1 + nx + [n(n − 1)/2]x².
Sum of the coefficients: put x = 1.

Start here: Expanding powers of a bracket: spotting the pattern

You already know (a + b)² = a² + 2ab + b². What about (a + b)³ or (a + b)¹⁰? Multiplying out ten brackets by hand would take ages. Mathematicians noticed a beautiful pattern that lets us skip all that work. Let us discover it ourselves, the way a detective gathers clues.

Worked example 8.1

Expand (a + b)³ by multiplication.

(a + b)³ = (a + b)(a + b)², and we know (a + b)² = a² + 2ab + b².
Multiply each term of (a² + 2ab + b²) by a: a³ + 2a²b + ab².
Multiply each term by b: a²b + 2ab² + b³.
Add, collecting like terms: a³ + 3a²b + 3ab² + b³.

a³ + 3a²b + 3ab² + b³

Now line up the results:

n	(a + b)ⁿ	Coefficients
1	a + b	1 1
2	a² + 2ab + b²	1 2 1
3	a³ + 3a²b + 3ab² + b³	1 3 3 1

Look closely, and you will see three clues:

Number of terms: (a + b)ⁿ has n + 1 terms.
Powers: the power of a starts at n and goes down by 1 each term, while the power of b starts at 0 and goes up by 1. In every term, the two powers add up to n (in 3a²b, 2 + 1 = 3).
Coefficients: they are symmetrical (1 3 3 1 reads the same backwards), and they follow a pattern we explore in 1.2.

⚠️ Common mistake: writing (a + b)² = a² + b². The middle term 2ab is missing! Test it with numbers: (2 + 3)² = 25, but 2² + 3² = 13.

Try it yourself. How many terms are in the expansion of (x + y)⁹, and what is the sum of the powers in each term?

Show solution

10 terms; the powers in each term add up to 9.

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Practice questions

1. Expand (1 − 2x)⁴.

Show worked answer

1 − 8x + 24x² − 32x³ + 16x⁴.

2. Find the 3rd term of (x + 3)⁵.

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⁵C₂ x³ 3² = 90x³.

3. Find the coefficient of x³ in (2 + x)⁶.

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20 × 8 = 160.

4. Find the term independent of x in (x² + 1/x)⁶.

Show worked answer

r = 4: 15.

5. Evaluate (1.01)⁶ to 4 d.p.

Show worked answer

1.0615.

Key words

Binomial: An expression with two terms, e.g. (x + 3).
Expansion: Writing a power of a bracket as a sum of terms.
Coefficient: The number multiplying the letters in a term.
Pascal's triangle: A triangle of numbers giving the binomial coefficients.
Factorial (n!): n × (n − 1) × … × 2 × 1; 0! = 1.
ⁿCᵣ: "n choose r" = n!/[r!(n − r)!], the number of ways to choose r items from n.
General term: The formula Tᵣ₊₁ for any term of the expansion.
Term independent of x: The constant term, where the power of x is 0.
Ascending powers: Powers of x increasing: 1, x, x², …

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