Mathematics (IGCSE, Grade 9) · Textbook Exercise 3.4, Q1–12

Math Exercise 3.4 — Factorising & Solving Cubics

Graded 2026-06-17 · Sys4Ethan (Claude Opus 4.8 vision, human-verified)

67%

✅ Correct 8 ◐ Partial 4 ✗ Needs fixing 0 ○ Not attempted 0 Total 12

To fix & review (4)Q8 Q9 Q10 Q11

Q1 ✓ Correct factor theorem + factorise

$x-1$ shown as a factor, then full factorisation $(x-1)(2x-1)(x+1)$.

Q2 ✓ Correct factorise cubics completely (a–h)

All eight parts factorised completely and correctly — excellent.

Q3 ✓ Correct solve cubic equations (a–h)

All eight solved; roots correct.

Q4 ✓ Correct roots in surd form (a–d)

Surd roots via the quadratic formula — all correct.

Q5 ✓ Correct surd roots

$x=-\tfrac12,\ -2\pm\sqrt{13}$.

Q6 ✓ Correct solve to 2 d.p.

$x=-3,\ 0.54,\ -5.54$.

Q7 ✓ Correct one real root argument

Good. For full marks, state explicitly that $x^2+x+1$ has no real roots (its discriminant is negative).

Q8 ◐ Partial form $f(x)$ from roots (coeff 1)

Try this → You've written down the given roots, but the question asks for $f(x)$. Build it: multiply the factors $(x-\text{root})$ together and expand to a cubic with integer coefficients (leading coefficient 1).

Q9 ◐ Partial form $f(x)$ from roots (coeff 2)

Try this → Same as Q8, but the leading coefficient is 2 — so use a factor like $(2x-1)$ for the root $\tfrac12$ before expanding.

Q10 ◐ Partial form $f(x)$ from a surd pair

Try this → The surd pair $1\pm\sqrt2$ comes from $x^2-2x-1$. Multiply that by $(x+3)$ and expand — your final cubic should be monic (leading coefficient 1). Please re-check your result.

Q11 ◐ Partial form $f(x)$ from a surd pair (coeff 2)

Try this → Method is right: $2\pm\sqrt3$ gives $x^2-4x+1$, times $(2x-1)$ for the root $\tfrac12$. Re-check the middle ($x$) term when you expand — it looks off.

Q12 ✓ Correct quartic factor theorem

Handled the quartic correctly: showed $4a^3-9a^2+4=0$ and got $a=2,\ \tfrac{1\pm\sqrt{33}}{8}$.

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