Mathematics (IGCSE, Grade 9) · Textbook Exercise 3.3, Q1–8

Math Exercise 3.3 — Factor Theorem & Polynomials

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

88%

✅ Correct 15 ◐ Partial 1 ✗ Needs fixing 0 ○ Not attempted 1 Total 17

To fix & review (2)Q2c Q7b

Q1a ✓ Correct factor theorem

$f(4)=0$ shown.

Q1b ✓ Correct factor theorem

Q1c ✓ Correct factor theorem

Q1d ✓ Correct factor theorem (root $-\tfrac13$)

Right root identified; for full marks substitute it and show the result is $0$.

Q2a ✓ Correct find unknown coefficient

$a=29$.

Q2b ✓ Correct find unknown coefficient

$a=546$.

Q2c ◐ Partial find unknown coefficient

Try this → Your setup $-\tfrac{27}{2}+\tfrac{9a}{4}-\tfrac{87}{2}+30=0$ is correct — you just stopped. Combine the number terms, then isolate $\tfrac{9a}{4}$ and solve for $a$.

Q3 ✓ Correct express $b$ in terms of $a$

$b=-2a-2$.

Q4a ✓ Correct quadratic factor → two conditions

Q4b ✓ Correct quadratic factor → two conditions

Q4c ✓ Correct quadratic factor → two conditions

All three of Q4 solved correctly — solid simultaneous-equation work.

Q5 ✓ Correct common factor

$a=-6$.

Q6 ✓ Correct common factor, two unknowns

$a=\tfrac{27}{2},\ b=\tfrac{11}{2}$ — nicely done.

Q7a ✓ Correct two factors → find $p,q$

$p=1,\ q=9$.

Q7b ○ Not attempted explain a third factor

Try this → Not answered. A cubic has three linear factors and you already have two of them — think about what the third must be (or test $x=-3$ in the expression and state the factor theorem).

Q8a ✓ Correct factor theorem → prove identity

$f(-a)=0$ gives $a^3-4a^2+3a=0$.

Q8b ✓ Correct solve the cubic in $a$

Correct: $a=0,1,3$. (Note $a=0$ makes the factor just $x$ — still valid.)

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