Binomial Expansion for Rational n

Which is the reason the expansion of $(1+x)^n$, for a rational non-integer $n$, never terminates the way $(1+x)^5$ does?

• The coefficient-generating product $n(n-1)(n-2)\cdots(n-r+1)$ never contains a zero factor for any $r$, because counting down from a non-integer $n$ in whole steps never lands exactly on zero — unlike counting down from a whole number like 5, which does.

  Correct. For $n=5$, the factor $(n-5)=0$ enters the product once $r$ reaches 6, and every later term is multiplied by that zero. For $n=\frac12$ or $n=-2$, no whole-step count-down from $n$ ever reaches exactly zero, so the product — and the series — never stops.

• BBecause $x$ is a variable rather than a fixed number.

  $x$ being a variable is true of every binomial expansion, finite or infinite — it has nothing to do with why some terminate and others don't. The termination question is entirely about whether $n$ is a non-negative integer.

• CBecause the formula in the booklet only applies to positive whole-number n, so a different, separately infinite formula has to be used instead.

  There is only one general formula, and it applies to every rational n, integer or not — the booklet's P4 section states it once, without restriction. It's the SAME formula in both cases; what changes is only whether the coefficient product it generates happens to hit a zero.

• DBecause the binomial coefficients $\binom{n}{r}$ don't exist for non-integer n.

  They do exist — the general term formula IS the extended definition of that coefficient, built from $n(n-1)\cdots(n-r+1)/r!$ rather than the factorial-ratio form that only makes sense for a non-negative integer $n$. Nothing breaks; the coefficient is simply computed a different way.

Traps tested: Irrelevant property cited · Single formula misread as two separate formulae · Extended coefficient assumed undefined

The first three terms of $(1-4x)^{-2}$ are required. A student sets $u=4x$ instead of the correct $u=-4x$. Which statement about the resulting error is true?

• Every ODD-power term (the $x^1$ term, the $x^3$ term, ...) comes out with the wrong sign, while every EVEN-power term (the constant, the $x^2$ term, ...) comes out numerically correct despite the mistake.

  Correct, and checkable directly: the correct expansion is $1+8x+48x^2+256x^3+\ldots$; with $u=+4x$ substituted instead of $u=-4x$, the $x^1$ term flips to $-8x$ and the $x^3$ term flips to $-256x^3$, but the $x^2$ term stays $48x^2$ in both cases — because raising $+4x$ or $-4x$ to an EVEN power gives the same result either way, while raising them to an ODD power doesn't.

• BOnly the constant term is affected.

  The constant term (the '1' at the very start) doesn't involve $u$ at all, in either version — it can't be affected by a sign error inside $u$. This is backwards: the constant is the one term that's guaranteed safe.

• CThe whole expansion comes out completely different, with no terms matching between the two versions.

  Too strong — every even-power term is identical between the two versions, which is exactly what makes this error dangerous rather than obvious: a student who only re-checks the first couple of terms (constant and $x^2$) can find them looking right and stop checking.

• DThere's no real difference, since raising a term to a power removes its sign either way.

  That's true only for EVEN powers — raising a number to an odd power keeps its sign. $(-4x)^1=-4x$ but $(4x)^1=4x$: not the same number. The claim only holds for half the terms.

Traps tested: Constant term assumed affected · Extent of error overstated · Even power sign behaviour overgeneralised

To expand $(1-5x^2)^{\frac14}$ using the general binomial series, what should be substituted for $u$?

• $u=-5x^2$ — the whole term multiplying the 1, $x^2$ included.

  Correct. The general formula doesn't care that $x$ is squared inside $u$ — it only needs $u$ itself to be small. Treating $-5x^2$ as a single quantity is exactly what makes the direct-expansion method in this lesson's method-comparison block work.

• B$u=-5x$ — drop the square, since the formula is 'really' about $x$.

  This is the exact, documented error: a real examiner report records candidates 'confused by the 4x² term, with some replacing it with 4x' on a structurally identical question. The formula has no opinion about what $u$ looks like algebraically — it works for $u=x$, $u=x^2$, $u=x^3$, or any other expression, as long as $|u|<1$.

• CYou can't apply the formula directly — you have to first write $1-5x^2=(1-\sqrt5x)(1+\sqrt5x)$ and expand each bracket separately.

  This route is mathematically valid but unnecessary — and, as the method-comparison block shows directly, considerably more work for the same answer. Presenting it as REQUIRED, rather than optional and slower, is itself part of the misconception.

• D$u=-25x^4$ — square the whole $-5x^2$ term before substituting.

  This squares $u$ before it's even been substituted into the formula, which then squares it again inside the $x^2$-coefficient term — $u$ itself should just be $-5x^2$, unsquared; the squaring happens automatically wherever the formula itself calls for $u^2$.

Traps tested: X squared term collapsed into x · Unnecessary factoring route assumed required · U pre squared before substitution

For the expansion of $(2+8x)^{-\frac13}$, written as $2^{-\frac13}(1+4x)^{-\frac13}$, what is the correct range of validity?

• $|x|<\frac14$

  Correct. Here $u=4x$, so the condition is $|4x|<1$, i.e. $|x|<\frac14$ — derived from $u$ directly, not read off the original coefficients 2 and 8 by pattern.

• B$|x|<4$

  This uses the coefficient of $x$ inside the bracket (4) directly as the bound, rather than its RECIPROCAL. A real report records this exact style of wrong answer — '|x| < 4' — on a structurally identical range-of-validity question (Oct 2022, Q4).

• C$x<\frac14$

  The bound is right and the modulus is missing — this only captures $x$ values up to $\frac14$ on the positive side, and says nothing about $x=-\frac12$, which is also outside the true range. A real report records exactly this style of error too, dropping the modulus bars from an otherwise-correct-looking bound (Oct 2022, Q4).

• D$|x|<\pm\frac14$

  This isn't meaningful notation — a modulus inequality already ranges over both positive and negative x, so adding a $\pm$ beside it doesn't specify anything different or extra. A real report records this exact confused form as a genuine, repeated wrong-answer pattern (Oct 2022, Q4), most likely from students trying to explicitly signal 'both signs' without realising the modulus already does that job on its own.

Traps tested: Range bound not inverted · Modulus dropped from validity range · Modulus and plus minus both used

The worked chain earlier in this lesson found $(9-6x)^{-\frac12} \approx \frac13+\frac19x+\frac{1}{18}x^2+\frac{5}{162}x^3$. Substituting $x=\frac{1}{10}$ gives an approximation to $\frac{1}{\sqrt{8.4}}$.

A question asks for this approximation "as an exact fraction." Which final line is safest?

• $\frac{11179}{32400}$ — the four terms added as fractions over a common denominator, left unevaluated as a decimal.

  Correct, and checkable: $\frac13+\frac{1}{90}+\frac{1}{1800}+\frac{5}{162000} = \frac{11179}{32400} \approx 0.345031$, which matches $\frac{1}{\sqrt{8.4}} \approx 0.345033$ to 3 s.f. — the exact fraction is the deliverable the question actually asked for.

• B$0.345$ — the exact fraction sum, converted to a 3-significant-figure decimal.

  This is the specific, documented error: converting an exact value to a rounded decimal when the question explicitly asked for an exact form. A real report records this directly — 'a very small minority of candidates unfortunately gave a decimal approximation for their answer' when an exact surd form was required (Jan 2021, Q1).

• C$\sqrt{8.4}$, rearranged algebraically without ever substituting a numeric value into the expansion.

  This never actually uses the expansion — it's a restatement of the target, not an approximation derived from the series. The whole point of the technique is producing a rational (fraction) value FROM the series that's close to this surd, not manipulating the surd itself.

• D$\frac13$ — just the constant term, since the remaining terms involving $\frac{1}{10}$ are 'too small to matter.'

  Dropping later terms without being told to is not the same as the question's own required precision — nothing in the question authorised stopping after one term, and $\frac13\approx0.333$ is not even correct to 2 s.f. against the true value of $\approx0.345$.

Traps tested: Decimal given when exact form required · Expansion not actually used · Higher order terms dropped without instruction

A question gives the expansion of $(1+2x)^{-1}$ and says: 'by substituting x = 0.01, find an approximation to $\frac{1}{1.02}$.' A student substitutes into the series only, and writes down the resulting decimal as the final answer with no further comment.

What, specifically, is put at risk by stopping there?

• The mark for identifying WHAT the computed number is an approximation of — a real report records scores of zero for exactly this pattern, substituting into only one side of the identity.

  Correct. Showing $1+2(0.01)=1.02$ — substituting the same x into the ORIGINAL, unexpanded expression — is what connects the series' decimal output back to $\frac{1}{1.02}$ specifically. Skipping it is treated, per a real report on this exact question type, as though the question hadn't actually been answered: 'scores of 0 marks were very common with x = ¼ being substituted into only one side of the expansion' (Oct 2021, Q4).

• BNothing — the numeric value is all the question asked for.

  This is the misconception the trap is built around: a bare decimal, with nothing connecting it back to the original expression, is not distinguishable from an approximation to a completely different number.

• CThe mark for using the correct value of n.

  n isn't in question here — the student's use of n inside the series substitution is exactly what's being described as correct ("substitutes into the series only"). The value at risk is a different, separate mark: the one for stating what target value the series was approximating.

• DThe mark for stating the range of validity.

  The range of validity isn't part of what this question asked for — it governs whether the expansion is trustworthy at all, which is a separate concern from identifying what a specific numeric substitution approximates.

Traps tested: Identification step treated as optional · Unrelated mark identified