Mathematics 9709/12 — May/June 2019

Walkthrough

We are asked for the coefficient of $x$ in the expansion of $\left(\frac{2}{x} - 3x\right)^5$. The binomial theorem tells us that every term in the expansion has the form

for some integer $r$ from $0$ to $5$. In this general term, the power of $x$ comes from two places: the $\frac{1}{x}$ factor in $\left(\frac{2}{x}\right)^{5-r}$ and the $x$ factor in $(-3x)^r$. Combining these gives the exponent $2r - 5$.

To find the coefficient of $x$, we need this exponent to equal $1$:

so $r = 3$. This means the term we need is the one with $\left(\frac{2}{x}\right)^2$ and $(-3x)^3$.

Now we evaluate that term:

The $x$ powers simplify to $x$, and the constant part is $10 \cdot 4 \cdot (-27) = -1080$. Therefore the coefficient is $-1080$.

Key Takeaways

This question tests the binomial expansion of an expression with two different powers of $x$. The key skill is writing down the general term, combining the powers of $x$ from both factors, and then choosing the value of $r$ that gives the required power. It also reinforces that the coefficient is the constant multiplier of the term, not the term including the variable.

Common Mistakes

• Choosing the wrong value of $r$. A common error is using $r = 2$, which gives the term in $x^{-1}$ instead of $x$.
• Forgetting the negative sign. Since the second term is $-3x$, the cube $(-3x)^3$ contributes a negative sign.
• Mixing up the coefficient and the full term. The coefficient is $-1080$, while the full term is $-1080x$.
• Incorrectly combining the powers of $x$. Remember that $x^{-(5-r)} \cdot x^r = x^{2r-5}$.

Things to Be Careful About

• The exponent of $x$ in the required term must be exactly $1$, so solve $2r - 5 = 1$ carefully.
• $r$ must be an integer between $0$ and $5$; here $r = 3$ is valid.
• The binomial coefficient $\binom{5}{3}$ equals $10$, and $\binom{5}{2}$ would also equal $10$, but only $r = 3$ gives the correct power of $x$.
• The mark scheme awards marks for identifying the correct term structure and then for simplifying to $-1080$; showing the full term $-1080x$ is also acceptable.