Sub-topic 1
2025 · National One-Eighth
Free
Cubic $x^3+px^2+qx+6$ divisible by $x-2$ and $x-1$: find $p$ and $q$
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Sub-topic 2
2025 · National One-Eighth
Premium
Factorising $x^3-3x^2+2x-6$ by grouping
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Sub-topic 3
2024 · National One-Eighth
Premium
Sum and difference of two cubes, such as $x^3-y^3=(x-y)(x^2-xy-y^2)$ (true/false)
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Sub-topic 4
2026 · National One-Eighth
Premium
Cube factorisations such as $8a^3+b^3=(2a+b)(4a^2-2ab+b^2)$ (true/false)
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Sub-topic 5
2023 · National Preliminary
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Cube factorisations such as $x^3+27=(x-3)(x^2+3x-9)$ (true/false)
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Sub-topic 6
2026 · National Preliminary
Premium
Integer root of $x^3-2x^2=75$
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Sub-topic 7
2026 · National Semifinal
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$ax^3+2x^2+bx-4$ divisible by $(x+1)^2$: find $a$ and $b$
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Sub-topic 8
2026 · National Quarterfinal
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Factorising $(2x+3y)^2-(3x-2y)^2$
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Sub-topic 9
2026 · National One-Eighth
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Difference of two squared binomials, $(2x+3y)^2-(3x-2y)^2$
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Sub-topic 10
2026 · National One-Eighth
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Constants in $a(x-3)^2+b(x-3)+c\equiv2x^2-5x-12$
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Sub-topic 11
2018 · National Preliminary
Premium
Solving $x^3-2x^2+x-2=0$ by grouping
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Sub-topic 12
2018 · National Quarterfinal
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Solving cubic equations such as $x^3+7x^2-x-7=0$
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Sub-topic 13
2021 · National Preliminary
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Difference of two squares factorisations such as $x^2-3=(x-\sqrt{3})(x+\sqrt{3})$ (true/false)
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Sub-topic 14
2021 · National Preliminary
Premium
Factor theorem: $x^2-25$ is a factor of $x^3-125$ (true/false)
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Sub-topic 15
2023 · National Preliminary
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Factor theorem: $(x+2)$ is a factor of $x^3+4x^2+6x-2$ (true/false)
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Sub-topic 16
2024 · National One-Eighth
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Factorising $x^3-3x^2+4$ completely
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Sub-topic 17
2024 · National Preliminary
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Factorising cubics such as $x^3+2x^2-x-2$ completely
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Sub-topic 18
2023 · National Preliminary
Premium
Remainder of $3x^3+5x^2-3x+4$ divided by $x+2$
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Sub-topic 19
2013 · National Preliminary
Premium
Factorising $49x^2-81y^2$
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Sub-topic 20
2013 · National Preliminary
Premium
Factorising $\dfrac{a^2}{4}-\dfrac{b^2}{9}$
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Sub-topic 21
2013 · National Preliminary
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Factorising $\dfrac{9x^2}{4}-\dfrac{16y^2}{25}$
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Sub-topic 22
2013 · National Preliminary
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Evaluating $121^2-21^2$
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Sub-topic 23
2013 · National Preliminary
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Evaluating $85^2-15^2$
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Sub-topic 24
2013 · National Preliminary
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Evaluating $73^2-23^2$
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Sub-topic 25
2016 · National One-Eighth
Premium
Remainder of $x^3-6x^2+3x-2$ divided by $x-2$
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Sub-topic 26
2016 · National One-Eighth
Premium
Remainder of $3x^3+10x^2-7x+6$ divided by $x+1$
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Sub-topic 27
2016 · National One-Eighth
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Remainder of $2x^3+7x^2+8x-16$ divided by $x-1$
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Sub-topic 28
2016 · National Preliminary
Premium
Simplifying $(x+y)^2-(x-y)^2$
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Sub-topic 29
2017 · National Semifinal
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Factorising $5x^2-4xy-12y^2$
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Sub-topic 30
2017 · National Quarterfinal
Premium
$x^3+ax^2-4x+5$ leaves remainder $-5$ on division by $x+1$: find $a$
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Sub-topic 31
2020 · National One-Eighth
Premium
Is $x-2$ a factor of $3x^3-5x^2-2x$? (true/false)
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Sub-topic 32
2020 · National One-Eighth
Premium
Is $x+2$ a factor of $2x^3+7x^2+3x+1$? (true/false)
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Sub-topic 33
2020 · National One-Eighth
Premium
Is $x-1$ a factor of $2x^3-9x^2+5x+2$? (true/false)
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Sub-topic 34
2021 · National Preliminary
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Finding $k$ when $x-2$ is a factor of $x^3-3x^2+2x+k$
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Sub-topic 35
2021 · National Preliminary
Premium
Remainder of $3x^3-4x^2+2x-10$ divided by $x-2$
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Sub-topic 36
2022 · National Preliminary
Premium
Factor theorem: $x-a$ a factor of $f(x)$ implies $f(a)=0$ (true/false)
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Sub-topic 37
2022 · National Preliminary
Premium
If $ax+b$ is a factor of $f(x)$ then $f(-b)=0$ (true/false)
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Sub-topic 38
2022 · National Preliminary
Premium
If $ax+b$ is a factor of $f(x)$ then $f\left(-\dfrac{b}{a}\right)=0$ (true/false)
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Sub-topic 39
2023 · National Preliminary
Premium
Remainder of $3x^3+5x^2-3x+4$ divided by $x+2$ using the remainder theorem
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