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PSLE

PSLE Foundation Mathematics

<ol class="i-list"><li data-list-item-id="amint-q03-question-1" data-list-item-path="amint-q03-question-1"><div class="list-item-paragraph">Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where&nbsp;$a$, $b$, $c$ and $d$ are integers.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q03-question-2" data-list-item-path="amint-q03-question-2"><div class="list-item-paragraph">Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.<span class="inline-marks">[4]</span></div></li></ol>

Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where $a$, $b$, $c$ and $d$ are integers.

Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.

Integration Using Partial Fractions

Integrals Involving Long Division

Partial Fractions

Polynomial Division and Remainders

Definite Integral

Improper rational definite integral

Integration

Polynomials, Cubic Equations and Partial Fractions

<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">Explain why $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ is considered an improper fraction.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Express $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ in the form of $A + \frac{Bx + C}{(x^2 + 4)(x - 2)}$ and hence, express $\frac{2x^3 - 4x^2 + 13x - 10}{(x^2 + 4)(x - 2)}$ in partial fractions.<span class="inline-marks">[6]</span></div></li></ol>

2024 CCHY P1 Q5

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Using long division, divide $2x^3 - 7x^2 + 2x + 3$ by $2x - 4$.</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Solve $2x^3 - 7x^2 + 2x + 3 = 0$.</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Kun Ye claims that the solutions to the equation $2x^3 - 7x^2 + 2x + 3 = 0$ can be used to solve $3y^6 + 2y^4 - 7y^2 + 2 = 0$. Explain how he is correct.</span><span class="inline-marks">[2]</span></div></li></ol>

Using long division, divide $2x^3-7x^2+2x+3$ by $2x-4$.

Kun Ye claims that the solutions to the equation $2x^3-7x^2+2x+3=0$ can be used to solve $3y^6+2y^4-7y^2+2=0$. Explain how he is correct.

Cubic Expressions and Equations

2025 DSS PRELIMS P2 Q8

<p>Express $\frac{2x^2-11x+12}{x^2-5x+6}$ in partial fractions. Hence evaluate $\displaystyle\int_4^5 \frac{2x^2-11x+12}{x^2-5x+6} \, \mathrm{d}x$.</p>

Express $\frac{2x^2-11x+12}{x^2-5x+6}$ in partial fractions. Hence evaluate $\displaystyle\displaystyle \displaystyle\int_4^5 \frac{2x^2-11x+12}{x^2-5x+6} \, \mathrm{d}x$.

<ol class="abc-list"><li><div class="list-item-paragraph">When a polynomial $\mathrm{f}\left( x \right)$ is divided by $x-2$ and $x-4$, the remainder is $-1$ and $7$ respectively. Find the remainder when&nbsp;$\mathrm{f}\left( x \right)$ is divided by $\left( x-2 \right)\left( x-4 \right)$.</div></li><li><div class="list-item-paragraph">Given that the polynomial ${{x}^{2}}-2x+6$ leaves the same remainder when it is divided by $x-p$ and $x+3q$, find the value of $p-3q$ when&nbsp;$p\ne -3q$.</div></li></ol>

Remainder and Factor Theorems

<ol class="abc-list"><li><div class="list-item-paragraph">Find $\mathrm{Q}\left( x \right)$ if&nbsp;${{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x+3=\left( {{x}^{2}}-5x+3 \right)\mathrm{Q}\left( x \right)+9$.</div></li><li><div class="list-item-paragraph">Solve ${{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x-6=0$. Hence sketch the curve $y={{x}^{4}}-6{{x}^{3}}+6{{x}^{2}}+7x-6$ and state the range of values of 	$x$ such that</div><ol class="i-list"><li><div class="list-item-paragraph">$y&gt;0$,</div></li><li><div class="list-item-paragraph">$\frac{1}{y}&lt;0$.</div></li></ol></li></ol>

Polynomial Inequalities and Sign Charts

Equations Reducible to Quadratics

Polynomial Functions and Key Features

Quadratic Functions, Equations and Inequalities

<p>It is given that&nbsp;$\mathrm{f}\left( x \right)={{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}-2x+1$ and $\mathrm{g}\left( x \right)=3{{x}^{4}}-48$, where&nbsp;$a$ and $b$ are integers.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">If&nbsp;$\mathrm{f}\left( x \right)$ has a remainder of $2x-3$ when it is divided by ${{x}^{2}}-4$, find the value of $a$ and of $b$.</div></li><li><div class="list-item-paragraph">Factorise $\mathrm{g}\left( x \right)$ completely. Hence state the real values of $x$ if&nbsp;$\mathrm{g}\left( x \right)=0$.</div></li><li><div class="list-item-paragraph">Given further that&nbsp;$2\mathrm{f}\left( x \right)-3\mathrm{g}\left( x \right)=\mathrm{Q}\left( x \right)\left( x+2 \right)+\mathrm{R}\left( x \right)$, where&nbsp;$\mathrm{Q}\left( x \right)$ and $\mathrm{R}\left( x \right)$ are the quotient and remainder 	respectively, find $\mathrm{R}\left( x \right)$.</div></li></ol>

It is given that $\mathrm{f}\left( x \right)={{x}^{4}}+a{{x}^{3}}+b{{x}^{2}}-2x+1$ and $\mathrm{g}\left( x \right)=3{{x}^{4}}-48$, where $a$ and $b$ are integers.

Factorisation by Special Identities

<p>Without doing the long division, find the remainder when $3{{x}^{4}}-5{{x}^{2}}+4$ is divided by ${{x}^{2}}+2$.</p>

Without doing the long division, find the remainder when $3{{x}^{4}}-5{{x}^{2}}+4$ is divided by ${{x}^{2}}+2$.

<p>Express $\frac{8{{x}^{3}}+8{{x}^{2}}+4x-5}{4{{x}^{2}}+1}$ as the sum of a polynomial and a proper fraction.</p>

Express $\frac{8{{x}^{3}}+8{{x}^{2}}+4x-5}{4{{x}^{2}}+1}$ as the sum of a polynomial and a proper fraction.

<p>Given that $2{{x}^{3}}+{{x}^{2}}-3x-4\equiv (2x+A)(x+1)(x-3)+B(x+1)+C$ for all values of $x$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of $A$, $B$ and $C$.</div></li><li><div class="list-item-paragraph">Hence, deduce the remainder when $2{{x}^{3}}+{{x}^{2}}-3x-4$ is divided by ${{x}^{2}}-2x-3$.</div></li></ol>

Given that $2{{x}^{3}}+{{x}^{2}}-3x-4\equiv (2x+A)(x+1)(x-3)+B(x+1)+C$ for all values of $x$.

Comparing Coefficients in Polynomial Identities

<p>Given that $12{{x}^{3}}-61{{x}^{2}}+4x+16\equiv (Ax+B)(x-1)(x+3)+C(x-1)+D$ for all values of $x$, find the values of $A$, $B$, $C$ and $D$. Hence, or otherwise, deduce the remainder when $12{{x}^{3}}-61{{x}^{2}}+4x+16$ is divided by${{x}^{2}}+2x-3$.</p>

Given that $12{{x}^{3}}-61{{x}^{2}}+4x+16\equiv (Ax+B)(x-1)(x+3)+C(x-1)+D$ for all values of $x$, find the values of $A$, $B$, $C$ and $D$. Hence, or otherwise, deduce the remainder when $12{{x}^{3}}-61{{x}^{2}}+4x+16$ is divided by${{x}^{2}}+2x-3$.

<p>Given that&nbsp;$5{{x}^{3}}-17{{x}^{2}}+12x-6=({{x}^{2}}-3x+1)(5x+A)+Bx+C$ for all values of $x$, find the value of $A$, of $B$ and of $C$. Hence, find the remainder when&nbsp;$5x^3-17x^2+12x-6$ is divided by $({{x}^{2}}-3x+1)$.</p>

Given that $5{{x}^{3}}-17{{x}^{2}}+12x-6=({{x}^{2}}-3x+1)(5x+A)+Bx+C$ for all values of $x$, find the value of $A$, of $B$ and of $C$. Hence, find the remainder when $5x^3-17x^2+12x-6$ is divided by $({{x}^{2}}-3x+1)$.

<p>Find the quotient and remainder when&nbsp;$2{{x}^{3}}+7{{x}^{2}}-4x+15$ is divided by $2x+3$.</p>

Find the quotient and remainder when $2{{x}^{3}}+7{{x}^{2}}-4x+15$ is divided by $2x+3$.

<p>Divide $2{{x}^{3}}-{{x}^{2}}-7x-5$ by $2-x$ and state the quotient and remainder.</p>

Divide $2{{x}^{3}}-{{x}^{2}}-7x-5$ by $2-x$ and state the quotient and remainder.

<p>Divide $5{{x}^{3}}+6{{x}^{2}}+x-5$ by $x+1$ and state the quotient and remainder.</p>

Divide $5{{x}^{3}}+6{{x}^{2}}+x-5$ by $x+1$ and state the quotient and remainder.

<p>Divide $3{{x}^{4}}-{{x}^{3}}+2x+1$ by $x-1$ and state the quotient and remainder.</p>

Divide $3{{x}^{4}}-{{x}^{3}}+2x+1$ by $x-1$ and state the quotient and remainder.

Long Division of Polynomials

<p>Given that&nbsp;$3{{x}^{3}}+5{{x}^{2}}-4x-6\equiv \left( 3x+A \right)\left( x-1 \right)\left( x+4 \right)+B\left( x-1 \right)+C$ for all values of $x$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of $A$, $B$, and $C$.</div></li><li><div class="list-item-paragraph">Hence, deduce the remainder when&nbsp;$3{{x}^{3}}+5{{x}^{2}}-4x-6$ is divided by ${{x}^{2}}+3x-4$.</div></li></ol>

Given that $3{{x}^{3}}+5{{x}^{2}}-4x-6\equiv \left( 3x+A \right)\left( x-1 \right)\left( x+4 \right)+B\left( x-1 \right)+C$ for all values of $x$.

Hence, deduce the remainder when $3{{x}^{3}}+5{{x}^{2}}-4x-6$ is divided by ${{x}^{2}}+3x-4$.

<p>Given that&nbsp;$4{{x}^{3}}-3{{x}^{2}}-5x+2\equiv \left( Ax+B \right)\left( x-2 \right)\left( x+3 \right)+C\left( x-2 \right)+D$ for all values of $x$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the values of $A$, $B$, $C$, and $D$.</div></li><li><div class="list-item-paragraph">Hence, deduce the remainder when&nbsp;$4{{x}^{3}}-3{{x}^{2}}-5x+2$ is divided by ${{x}^{2}}+x-6$.</div></li></ol>

Given that $4{{x}^{3}}-3{{x}^{2}}-5x+2\equiv \left( Ax+B \right)\left( x-2 \right)\left( x+3 \right)+C\left( x-2 \right)+D$ for all values of $x$.

Find the values of $A$, $B$, $C$, and $D$.

Hence, deduce the remainder when $4{{x}^{3}}-3{{x}^{2}}-5x+2$ is divided by ${{x}^{2}}+x-6$.

<p>Given that&nbsp;$3{{x}^{3}}-8{{x}^{2}}+9x-7=\left( {{x}^{2}}-2x+3 \right)\left( 3x+A \right)+Bx+C$ for all values of $x$, find the values of $A$, $B$, and $C$. Hence, find the remainder when the expression is divided by $\left( {{x}^{2}}-2x+3 \right)$.</p>

Given that $3{{x}^{3}}-8{{x}^{2}}+9x-7=\left( {{x}^{2}}-2x+3 \right)\left( 3x+A \right)+Bx+C$ for all values of $x$, find the values of $A$, $B$, and $C$. Hence, find the remainder when the expression is divided by $\left( {{x}^{2}}-2x+3 \right)$.

<p>Given that&nbsp;$4{{x}^{3}}-7{{x}^{2}}+6x-3=\left( 2{{x}^{2}}-4x+1 \right)\left( 2x+A \right)+Bx+C$ for all values of $x$, find the values of $A$, $B$, and $C$. Hence, find the remainder when the expression is divided by $\left( 2{{x}^{2}}-4x+1 \right)$.</p>

Given that $4{{x}^{3}}-7{{x}^{2}}+6x-3=\left( 2{{x}^{2}}-4x+1 \right)\left( 2x+A \right)+Bx+C$ for all values of $x$, find the values of $A$, $B$, and $C$. Hence, find the remainder when the expression is divided by $\left( 2{{x}^{2}}-4x+1 \right)$.

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