8.4PureStretch

Solving Binomial Problems

Exam questions rarely ask for a whole expansion — they want one specific term or coefficient, or use a known coefficient to find an unknown. The general-term formula lets you jump straight to the part you need.

30 min Video by Zeeshan Zamurred The Binomial Expansion

Edexcel AS Level Maths: 8.4 Solving Binomial ProblemsWatch the full walkthrough before the notes below.

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What you'll be able to do

Find a specific term using the general term
Find the coefficient of a given power of x
Use a known coefficient to find an unknown constant
Avoid expanding more than necessary

The general term

Any single term of $(a + b)^{n}$ is given by one formula. Pick the value of $r$ that produces the power you want, and you never need the full expansion.

term = (r n) a^{n - r} b^{r}

Choose

r

to match the power of

x

you are after.

1Here

a = 1

b = x

n = 7

; take

r = 3

2Coefficient is

(3 7) = 35

Answer

35

Tip — Match the power first: decide which r gives the x-power you need, then compute just that term.

Terms with coefficients

When the bracket is like $(2 + 3 x)^{n}$ , substitute the full terms into the general term, including their powers, and simplify the number part.

1General term:

(r 5) 2^{5 - r} (3 x)^{r}

. For

x^{2}

r = 2

(2 5) 2^{3} (3)^{2} = 10 \times 8 \times 9

Answer

720

Finding an unknown

If a coefficient is given, set your expression for that coefficient equal to the given value and solve for the unknown constant. This turns a binomial fact into an equation.

1Coefficient of

x

(1 4) a = 4 a

2Set

4 a = 12

Answer

a = 3

Formula recap

(r n) a^{n - r} b^{r}

The general term.

r = power of the second term wanted

Pick r to match.

coefficient = given value \Rightarrow solve

Finding unknowns.

Common mistakes to avoid

Expanding the whole bracket to find one coefficient.

Use the general term and the right r — much faster and safer.

Forgetting the coefficient’s power, e.g. (3x)² gives 9x² not 3x².

Apply r to the whole second term, coefficient included.

Key takeaways

The general term nCr · aⁿ⁻ʳ · bʳ gives any single term.
Choose r to match the power of x you want.
Substitute full terms (with their powers) when there are coefficients.
For unknowns, set the coefficient expression equal to the given value.

Test yourself

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