A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

Show that $x - 1$ is a factor of $3x^3 + 8x^2 - 5x - 6$.

Hence solve the equation $3x^3 + 8x^2 - 5x = 6$ completely.

Hence solve the equation $3(y - 2)^3 + 8(y - 2)^2 - 5y + 4 = 0$.

Cubic Expressions and Equations

Remainder and Factor Theorems

2025 CGS PRELIMS P2 Q2

Polynomials, Cubic Equations and Partial Fractions

<p>The polynomial $\mathrm{P}(x)=x^3+ax^2-5x+6$ is divisible by $x-2$. Find $a$ and solve $\mathrm{P}(x)=0$.</p>

The polynomial $\mathrm{P}(x)=x^3+ax^2-5x+6$ is divisible by $x-2$. Find $a$ and solve $\mathrm{P}(x)=0$.

Unknown coefficient and remaining roots

<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Given that the expression $2x^3 + ax^2 + bx + c$ leaves the same remainder when divided by $(x - 5)$ and when divided by $(2x + 1)$, show that&nbsp;$9a + 2b = -91$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Given that the remainder is $-6$ when the expression $2x^3 + ax^2 + bx + c$ is divided by $(x - 1)$, and $(x + 1)$ is a factor, calculate the value of $c$.<span class="inline-marks">[4]</span></div></li></ol>

2024 GESS PRELIMS P2 Q1

<p>The polynomial $\mathrm{f}(x)$ is such that&nbsp;$\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where&nbsp;$a$ and $b$ are integers. It is given that&nbsp;$\mathrm{f}(x)$ is divisible by $2x - 3$ and that&nbsp;$\mathrm{f}'(1) = 6$.</p>
<ol class="abc-list"><li data-list-item-id="item-1"><div class="list-item-paragraph">Find the values of $a$ and $b$.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-2"><div class="list-item-paragraph">Using your values of $a$ and $b$, solve the equation $\mathrm{f}(x) = 0$.<span class="inline-marks">[3]</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ is such that $\mathrm{f}(x) = 6x^3 + ax^2 - 50x + b$, where $a$ and $b$ are integers. It is given that $\mathrm{f}(x)$ is divisible by $2x - 3$ and that $\mathrm{f}'(1) = 6$.

Power Rule and Derivatives of Algebraic Functions

2024 FMSS PRELIMS P1 Q7

Differentiation

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Given that&nbsp;$6x^2 - 9x + 18 = A(2x + 1)(x - 1) + B(x - 2) + C$ for all values of $x$, find the values of $A$, $B$ and $C$.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that&nbsp;$ \mathrm{f}(x) = 2x^3 - 11x^2 + 3x + 36$, show that&nbsp;$2x + 3$ is a factor of $ \mathrm{f}(x)$ and hence solve the equation $ \mathrm{f}(x) = 0$.<span class="inline-marks">[5]</span></div></li></ol>

Comparing Coefficients in Polynomial Identities

2024 CCHY P1 Q6

<p>It is given that&nbsp;$3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;$3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Solve $3p^3+7p^2-22p-8=0$.</span><span class="inline-marks">[6]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve $3p^3-7p^2-22p+8=0$.</span><span class="inline-marks">[2]</span></div></li></ol>

It is given that $3x^3-21x^2-22x+24$ when divided by $x+p$ has a remainder that is half the remainder when it is divided by $x-p$.

<ol class="num-list" start="4"><li data-list-item-id="item-4"><ol class="abc-list"><li data-list-item-id="item-4a"><div class="list-item-content"><span class="list-item-text">Find the remainder when&nbsp;$4x^3 + 3x^2 + 11x - 3$ is divided by $x + 2$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-4b"><div class="list-item-content"><span class="list-item-text">When a polynomial $g(x)$ is divided by $x - 1$ and $2x - 1$, the remainders are $1$ and $2$ respectively. Find the remainder when&nbsp;$g(x)$ is divided by $(x - 1)(2x - 1)$.</span><span class="inline-marks">[4]</span></div></li></ol></li></ol>

YTSS 4AM 2024 Prelims P1 Q4

<ol class="num-list" start="2"><li><div class="list-item-paragraph">The polynomial $P(x) = ax^3 + bx^2 - 19x + 4$, where&nbsp;$a$ and $b$ are constants, has a factor $x + 4$ and is such that&nbsp;$2P(1) = 5P(0)$.</div><ol class="abc-list"><li><div class="list-item-paragraph">Find the value of $a$ and of $b$.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence, factorise $P(x)$ completely.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

CGSS 4AM 2021 Prelims P1 Q2

<ol class="i-list" start="4"><li data-list-item-id="item-1776339941769-279hfobu7"><div class="list-item-paragraph">The equation of a polynomial is given by $\mathrm{f}(x) = 3x^3 - 11x^2 + 7x + 1$.</div><ol class="abc-list"><li data-list-item-id="subitem-1776339957417-40tyqkyyt"><div class="list-item-paragraph">Find the remainder when&nbsp;$\mathrm{f}(x)$ is divided by $x + 2$.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="subitem-1776339958325-ihojvef4w"><div class="list-item-paragraph">Solve the equation $3x^3 - 11x^2 + 7x + 1 = 0$.<span class="inline-marks">[7]</span></div></li></ol></li></ol>

SGSS 4AM Prelims P2 2022 Q4

<p>The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down a factor of $\mathrm{f}(x)$.</span></div></li></ol>
<p>It is further given that&nbsp;$\mathrm{f}'(a)=0$.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">What can be deduced about the factorised form of $\mathrm{f}(x)$?</span></div></li></ol>

The polynomial $\mathrm{f}(x)$ has the property $\mathrm{f}(a)=0$.

Write down a factor of $\mathrm{f}(x)$.

It is further given that $\mathrm{f}'(a)=0$.

What can be deduced about the factorised form of $\mathrm{f}(x)$?

Product Rule

NJC IP4 Techniques of Differentiation Tutorial Q4

<p>It is given that&nbsp;$\mathrm{f}(x) = 3x^3 + ax^2 - 7x + 3$, where&nbsp;$a$ is a constant, has a factor of $x + 1$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Find the value of $a$ and factorise $\mathrm{f}(x)$ completely.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b"><div class="list-item-content"><span class="list-item-text">Solve the equation $\mathrm{f}(\cot^2 \theta) = 0$ for&nbsp;$-90{}^\circ &lt; \theta &lt; 90{}^\circ$.</span><span class="inline-marks">[3]</span></div></li></ol>

It is given that $\mathrm{f}(x)=3x^3+ax^2-7x+3$, where $a$ is a constant, has a factor of $x+1$.

Find the value of $a$ and factorise $\mathrm{f}(x)$ completely.

Solve the equation $\mathrm{f}(\cot^2 \theta)=0$ for $-90^\circ < \theta < 90^\circ$.

Trigonometric Equations

(A Math) 2025 CCHY PRELIMS P2 Q4

Trigonometry

<p>The expression $2x^3 + ax^2 + bx + 10$, where&nbsp;$a$ and $b$ are constants, has a factor of $x + 2$ and leaves a remainder of $10$ when divided by $x - 3$. Find the values of $a$ and of $b$.<span class="inline-marks">[4]</span></p>

The expression $2x^3+ax^2+bx+10$, where $a$ and $b$ are constants, has a factor of $x+2$ and leaves a remainder of 10 when divided by $x-3$. Find the values of $a$ and of $b$.

2025 DSS PRELIMS P1 Q8

<p>When&nbsp;$f(x)$ is divided by $x-1$, the remainder is $61$. The remainder is $-123$ when&nbsp;$f(x)$ is divided by $x+3$. Find the remainder when&nbsp;$f(x)$ is divided by $x^2+2x-3$.<span class="inline-marks">[5]</span></p>

When $f(x)$ is divided by $x-1$, the remainder is $61$. The remainder is $-123$ when $f(x)$ is divided by $x+3$. Find the remainder when $f(x)$ is divided by $x^2+2x-3$.

2025 CGSS PRELIMS P1 Q5

<p>The cubic polynomial $\mathrm{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\mathrm{f}(x)=0$ are $-1$, $3$ and $k$. Given that&nbsp;$\mathrm{f}(x)$ has a remainder of $20$ when divided by $x-4$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$ ;</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">the remainder when&nbsp;$\mathrm{f}(x)$ is divided by $x+2$.<span class="inline-marks">[2]</span></div></li></ol>

The cubic polynomial $\mathrm{f}(x)$ is such that the coefficient of ${{x}^{3}}$ is $2$ and that the roots of the equation $\mathrm{f}(x)=0$ are $-1$, $3$ and $k$. Given that $\mathrm{f}(x)$ has a remainder of $20$ when divided by $x-4$, find

Forming Polynomials from Roots and Factors

IP3 Class Test (May 2025) Q6

<p>The function f is defined by $f(x) = 2x^3 + (4 - 2a)x^2 - ax + 6a$, where&nbsp;$a$ is a constant.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that&nbsp;$x + 2$ is a factor of $f(x)$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the other quadratic factor of $f(x)$ in terms of $a$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">If&nbsp;$f(x) = 0$ has only one real root, find the range of values of $a$.<span class="inline-marks">[2]</span></div></li></ol>

The function f is defined by $f(x) = 2x^3 + (4 - 2a)x^2 - ax + 6a$, where $a$ is a constant.

Show that $x + 2$ is a factor of $f(x)$.

Find the other quadratic factor of $f(x)$ in terms of $a$.

If $f(x) = 0$ has only one real root, find the range of values of $a$.

IP3 Class Test (May 2025) Q5

<p>The equation of a polynomial is given by $p(x) = x^3 - 7x^2 + 18x - 24$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that $x - 4$ is a factor of $p(x)$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the remainder when $p(x)$ is divided by $2x - 1$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that $p(x) = 0$ only has one real root.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Using the answers from (a) and (c), solve $2^{3y} - 7(2^{2y}) + 18(2^y) = 24$.</span><span class="inline-marks">[2]</span></div></li></ol>

The equation of a polynomial is given by $p(x) = x^3 - 7x^2 + 18x - 24$.

Show that $x - 4$ is a factor of $p(x)$.

Find the remainder when $p(x)$ is divided by $2x - 1$.

Show that $p(x) = 0$ only has one real root.

Using the answers from (a) and (c), solve $2^{3y} - 7(2^{2y}) + 18(2^y) = 24$.

2023 CSS AM S3 EOY Q6

<p>The polynomial $g(x) = mx^3 - 8x^2 - 9x + n$ has a factor of $3x - 2$ and leaves a remainder of $6$ when divided by $x$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the values of $m$ and $n$.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Using the values of $m$ and $n$ from <strong>(a)</strong>, solve, in exact form, the equation $g(x) = 0$.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence solve the equation $9e^{2y} + 8e^y = 12 + 6e^{3y}$.</span><span class="inline-marks">[3]</span></div></li></ol>

The polynomial $g(x) = mx^3 - 8x^2 - 9x + n$ has a factor of $3x - 2$ and leaves a remainder of $6$ when divided by $x$.

Hence solve the equation $9e^{2y} + 8e^y = 12 + 6e^{3y}$.

2023 Ahmad Ibrahim Sec AM S3 EOY Q9

<p>The function $\mathrm{f}(x) = 2x^3 + ax^2 + bx + 60$, where $a$ and $b$ are constants, is exactly divisible by $x + 4$. The remainder is $-18$ when $\mathrm{f}(x)$ is divided by $x - 2$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $a$ and of $b$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">By showing your working clearly, solve $\mathrm{f}(x) = 0$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence solve the equation $2(10^{6y}) + a(10^{4y}) + b(10^{2y}) = -60$, where $a$ and $b$ are the values found in part <strong>(a)</strong>.<span class="inline-marks">[3]</span></div></li></ol>

The function $\mathrm{f}(x) = 2x^3 + ax^2 + bx + 60$, where $a$ and $b$ are constants, is exactly divisible by $x + 4$. The remainder is $-18$ when $\mathrm{f}(x)$ is divided by $x - 2$.

By showing your working clearly, solve $\mathrm{f}(x) = 0$.

2023 ACSBR AM S3 EOY Q12 

<p>Find the value of $k$ for which $x-2$ is a factor of $\mathrm{f}(x)=3{{x}^{3}}-2{{x}^{2}}+5x+k$.</p>

Find the value of $k$ for which $x-2$ is a factor of $\mathrm{f}(x)=3{{x}^{3}}-2{{x}^{2}}+5x+k$.

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Given&nbsp;$\mathrm{f}(x) = 2x^3 + 3x^2 - 11x - 6$, find the remainder when&nbsp;$\mathrm{f}(x)$ is divided by $2x + 5$.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the remainder when&nbsp;$x^3 + 3x - 2$ is divided by $x + 2$.</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find the remainder when&nbsp;$(x + 2)^3(x - 1) - 6x + 3$ is divided by $x + 3$.</div></li></ol>

Given $\mathrm{f}(x) = 2x^3 + 3x^2 - 11x - 6$, find the remainder when $\mathrm{f}(x)$ is divided by $2x + 5$.

Find the remainder when $x^3 + 3x - 2$ is divided by $x + 2$.

Find the remainder when $(x + 2)^3(x - 1) - 6x + 3$ is divided by $x + 3$.

Math Question Bank