Binomial Expansion
Algebra · Binomial Expansion
Syllabus tag: KCSE | Mathematics | Form 3 | Topic 11 Binomial Expansion
Lesson objectives
By the end of this topic, you should be able to:
- Expand binomial expressions up to the power of four by multiplication.
- Build Pascal's Triangle up to the eleventh row.
- Use Pascal's Triangle to expand binomial expressions.
- Apply binomial expansion to numerical approximation.
Binomial Expansion
A binomial is an expression with two terms, such as a + b. This topic expands its powers without multiplying out every time.
a) By repeated multiplication
(a + b)¹ = a + b.
(a + b)² = a² + 2ab + b².
(a + b)³ = a³ + 3a²b + 3ab² + b³.
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴.
Multiplying out works but becomes slow. Look at the coefficients. They run 1 1, then 1 2 1, then 1 3 3 1, then 1 4 6 4 1.
b) Pascal's Triangle
Each row begins and ends with 1.
Every other entry is the sum of the two entries above it.
So row 5 is 1, 5, 10, 10, 5, 1. Check: 1+4 = 5, 4+6 = 10, 6+4 = 10, 4+1 = 5.
Continuing gives row 6 as 1, 6, 15, 20, 15, 6, 1.
The row number matches the power. Row 4 gives the coefficients for (a + b)⁴.
c) Using the triangle
Take the coefficients from the matching row.
The powers of the first term descend from n to 0.
The powers of the second term ascend from 0 to n.
Each term's two powers always add to n. That is a quick check on your working.
The expansion of (a + b)ⁿ always has n + 1 terms.
d) When terms are not simply a and b
Substitute carefully, keeping brackets.
Take (2x + 3)³. The coefficients are 1, 3, 3, 1. The terms are (2x)³, (2x)²(3), (2x)(3)² and (3)³.
That gives 8x³ + 36x² + 54x + 27.
The commonest error is forgetting to raise the coefficient inside the bracket. (2x)³ is 8x³, not 2x³.
e) With a negative second term
For (a − b)ⁿ, treat it as (a + (−b))ⁿ.
The signs then alternate: plus, minus, plus, minus.
So (a − b)³ = a³ − 3a²b + 3ab² − b³.
f) Numerical approximation
Binomial expansion evaluates awkward powers quickly.
For 1.02⁴, write it as (1 + 0.02)⁴.
Expanding gives 1 + 4(0.02) + 6(0.0004) + 4(0.000008) + …
That comes to 1 + 0.08 + 0.0024 + 0.000032. So 1.02⁴ is about 1.0824.
Later terms shrink rapidly when the second term is small. Two or three terms usually give enough accuracy. State how many you used.
g) Where this is used
Approximating powers without a calculator. Probability, where the coefficients count combinations. Compound interest over several periods.
Words to know
- Binomial -- an algebraic expression with exactly two terms.
- Pascal's Triangle -- the triangular array of binomial coefficients.
- Coefficient -- the numerical multiplier of a term in an expansion.
- Expansion -- the result of multiplying out a bracketed expression.
- Approximation -- an estimate obtained by using only the leading terms.
:::checkpoint Check yourself
- Write down row 5 of Pascal's Triangle.
- Expand (a + b)⁴.
- Expand (x + 2)³.
- How many terms are there in the expansion of (a + b)⁷? :::
Bridge to practice
The exercises begin with building Pascal's Triangle and expanding simple binomials, move through negative terms and coefficients, and finish with particular terms and numerical approximation. After every expansion, count the terms and check that each term's powers add to n.