Year 11 / Binomial expansion and cubic functions

Count the coefficients.

In (x + y)ⁿ, each factor contributes either x or y. Choosing y from r of the n factors gives the coefficient nCr. Its term is nCr × $x^{n}$ ⁻ʳyʳ. All terms have total degree n.

A minus sign belongs to the entire selected term. In ( $2x$ − 1)ⁿ, powers of −1 alternate signs, while powers of 2 also change the coefficients. A combination counts selections where order does not matter.

INTERACTIVE MODEL

Try it. Watch it change.

Power n8

(x + 1)ⁿ coefficients: 1, 8, 28, 56, 70, 56, 28, 8, 1
Selected r = 1: nCr = 8

Each coefficient counts selections of r factors that contribute 1 rather than x.

Explore: Change n and watch Pascal’s row become the coefficients of (x + 1)ⁿ.

Binomial theorem

(x+y)^n=\sum_{r=0}^n\binom nr x^{n-r}y^r

WORKED EXAMPLE

Find the coefficient of $x^{2}$ in (x + 3)⁴.

Use r = 2 so the power of x is 4 − 2 = 2.
The coefficient is 4C2 × $3^{2}$ = 6 × 9 = 54.

Assumed knowledge

Multiplication of polynomials and factorial notation.

Learning checkpoints & source

YOUR LEARNING CHECKPOINT

Use combinations and Pascal’s triangle.
Find a term in a binomial expansion.

QCAA Mathematical Methods 2025 v1.3 · p. 16 ↗

READY TO TRY IT?

Count the coefficients.

Find the coefficient of x2x^{2}x2 in (x + 3)⁴.

Put the idea to work.

Find the coefficient of $x^{2}$ in (x + 3)⁴.