Binomial Expansion

For positive integer $n$:

$(a + b)^n = \sum \binom{n}{r} a^{n-r} b^r$ ($r = 0$ to $n$)

with nCr from Pascal’s triangle or the formula. Full expansions are warm-ups; the marks live in specific terms, found with the general term $\binom{n}{r} a^{n-r} b^r$.

The general-term method

Find the coefficient of $x^3$ in $(2 - 3x)^5$. General term: $\binom{5}{r} \cdot 2^{5-r} \cdot$ $(-3x)^r$ ← the sign stays welded to its term $x^3$ requires $r = 3$: $\binom{5}{3} \cdot 2^2 \cdot (-3)^3 = 10 \cdot 4 \cdot (-27) =$ $-1080$

Set up the general term (M), identify $r$ from the power condition (M), evaluate with signs (A). The decisive habit: keep the entire second term, coefficient, sign, $x$, inside the bracket with its power. Splitting 3 from $x$ and patching the sign later is where this subtopic dies.

Read precisely what’s asked: the coefficient is $-1080$; the term is $-1080x^3$, they’re different answers to different commands.

Term independent of x

The signature hard variant uses brackets whose parts have opposing powers:

The term independent of $x$ in $\left(x^2 + \dfrac{2}{x}\right)^9$: General term: $\binom{9}{r} (x^2)^{9-r} \left(\dfrac{2}{x}\right)^r = \binom{9}{r} \cdot 2^r \cdot x^{18-2r-r}$ Independent of $x \Rightarrow 18 - 3r = 0 \Rightarrow r = 6$ $\binom{9}{6} \cdot 2^6 = 84 \cdot 64 =$ $5376$

The power-condition equation ($18 - 3r = 0$) is the method mark, write it as an equation, not a mental hop.

Two-bracket products

“Find the coefficient of $x^2$ in $(1 + 2x)(1 - x)^6$”: the $x^2$ term collects two contributions. $1 \times$ ($x^2$ term of the expansion) $+ 2x \times$ ($x^1$ term). Compute the needed expansion terms first, list both products, add. One contribution found = half the marks lost; the question is designed to test whether you see both.

Common mistakes

• The minus sign divorced from its term
• Coefficient vs term confused in the final line
• Power condition solved mentally (and wrongly) instead of as an equation
• Two-bracket questions answered with one contribution
• nCr values slipped (Pascal’s row written wrongly under time pressure, the formula is the check)

Full topic context: Series notes.