Mathematics (IGCSE, Grade 9) · Textbook Exercise 3.5, Q1–16

Math Exercise 3.5 — Remainder Theorem

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

88%

✅ Correct 14 ◐ Partial 1 ✗ Needs fixing 0 ○ Not attempted 1 Total 16

To fix & review (2)Q1 Q16

Q1 ◐ Partial find the remainder (a–d)

Parts (a), (c), (d) correct ($5,\ 76,\ 2$).

Try this → For (b), re-add $8-24+22-7$ carefully — group the positives and negatives and re-check that remainder.

Q2 ✓ Correct remainder → find a constant

$a=5,\ b=57,\ c=11$.

Q3 ✓ Correct factor + remainder conditions

$a=4,\ b=0$.

Q4 ✓ Correct factor + remainder conditions

$a=-6,\ b=-6$.

Q5 ✓ Correct find a,b then solve

$a=-8,\ b=15$, then solved $f(x)=0$ with surd roots.

Q6 ✓ Correct find a,b then a remainder

$a=-9,\ b=2$; remainder $5$.

Q7 ✓ Correct two remainder conditions

Showed $a=-4,\ b=2$.

Q8 ✓ Correct two remainder conditions

$a=6,\ b=-3$.

Q9 ✓ Correct express b in a, then solve

$b=12-2a$, then $a=5,\ b=2$.

Q10 ✓ Correct roots + remainder

$k=5$; remainder $-72$.

Q11 ✓ Correct factor + remainder conditions

$a=-8,\ b=-5$; remainder $-30$.

Q12 ✓ Correct R and 4R relation

$k=32$.

Q13 ✓ Correct R and 2R; prove + solve

Showed $3a^3-2a^2-18a-9=0$ and solved it: $a=3,\ \tfrac{-7\pm\sqrt{13}}{6}$.

Q14 ✓ Correct R and −R; then a remainder

$k=3$; remainder $-5$.

Q15 ✓ Correct nested-form remainders

$a=2,\ b=-5,\ c=7$ — clean use of the nested form.

Q16 ○ Not attempted challenge: roots 1, k, k+1

Try this → The challenge question — give it a go. With roots $1,k,k+1$ and $f(2)=20$, you can reach $k^2-3k-18=0$; then solve that quadratic for the possible $k$.

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