Worked example

Coefficient of x³ in (4−x)(1+3x)⁶

A-Level Maths · 97095 marksmed

Question

Find the coefficients of $x^{2}$ and $x^{3}$ in the expansion of $(1 + 3 x)^{6}$ . Hence find the coefficient of $x^{3}$ in the expansion of $(4 - x) (1 + 3 x)^{6}$ .

✓ Solved & verified by the OwlsPrep engine

Full worked solution

(i) Coefficient of $x^{2}$ in $(1 + 3 x)^{6}$
For a positive-integer binomial expansion of $(1 + u)^{n}$ , the general term with power $r$ of $x$ is $(r n) u^{r}$ , where $(r n) = \frac{n !}{r ! ( n - r )!}$ is the binomial coefficient. For the $x^{2}$ term, $r = 2$ , $n = 6$ , and $u = 3 x$ . Substitute into the formula:
$\text{Coefficient of } x^2 = \binom{6}{2} (3)^2$
Evaluate $(2 6) = 15$ and $3^{2} = 9$ , so $15 \times 9 = 135$ . Therefore the coefficient of $x^{2}$ in $(1 + 3 x)^{6}$ is $135$ .
Note.
- $(r n)$ counts the number of ways to pick $r$ copies of $u$ from $n$ factors
- Remember to raise the entire $u = 3 x$ term to the power $r$ , not just $x$
(ii) Coefficient of $x^{3}$ in $(1 + 3 x)^{6}$
Use the same positive-integer binomial term formula for $r = 3$ :
$\text{Coefficient of } x^3 = \binom{6}{3} (3)^3$
Evaluate $(3 6) = 20$ and $3^{3} = 27$ , so $20 \times 27 = 540$ . Therefore the coefficient of $x^{3}$ in $(1 + 3 x)^{6}$ is $540$ .
Note.
- You only need these two coefficients for the final step, no need to expand the entire polynomial
(iii) Coefficient of $x^{3}$ in $(4 - x) (1 + 3 x)^{6}$
An $x^{3}$ term in the product can only arise from two combinations: the constant term $4$ multiplied by the $x^{3}$ term of $(1 + 3 x)^{6}$ , and the $- x$ term multiplied by the $x^{2}$ term of $(1 + 3 x)^{6}$ . Combine these contributions:
$\text{Total coefficient} = 4 \times \left(\text{coeff of } x^3\right) + (-1) \times \left(\text{coeff of } x^2\right)$
Substitute the values from steps 1 and 2: $4 (540) - 1 (135) = 2160 - 135 = 2025$ . Therefore the coefficient of $x^{3}$ in $(4 - x) (1 + 3 x)^{6}$ is $2025$ .
Note.
- The negative sign from $- x$ applies to the entire second term; be careful not to drop it

Answer

Coefficient of $x^{2}$ : $135$ , Coefficient of $x^{3}$ in $(1 + 3 x)^{6}$ : $540$ , Final coefficient of $x^{3}$ : $2025$

What this tests

Positive-integer binomial expansion formula
Binomial coefficient calculation
Polynomial multiplication: combining like terms
Efficient partial expansion (only compute needed terms, not full polynomial)

⚠️

⚠ A common mistake is forgetting to raise the $3$ in $3 x$ to the power of $r$ , e.g. writing $(2 6) \cdot 3$ instead of $(2 6) \cdot 3^{2}$ , which gives an incorrectly small coefficient.

🔒

Master this question type — free

You’ve got the answer. A free account turns this one question into real exam readiness:

📝

Examiner mark schemeExactly where each mark is won — B1 / M1 / A1.

⚠️

Every common mistakeThe full list of traps that cost marks here.

💡

Exam techniqueFaster methods and what examiners reward.

🎯

15 more like thisA practice set tuned to your level.

🦉

Ask Ollie, the AI tutorStuck on a step? Get it explained a different way, instantly.

Create free account See how it works

Coefficient of x³ in (4−x)(1+3x)⁶

Full worked solution

What this tests

Master this question type — free

More from this topic