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PSLE

PSLE Foundation Mathematics

<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>

Cubic Expressions and Equations

Remainder and Factor Theorems

Comparing Coefficients in Polynomial Identities

2024 CCHY P1 Q6

Polynomials, Cubic Equations and Partial Fractions

<p>The sum of $\frac{3}{5x+4}$, $\frac{7}{4-5x}$ and $\frac{px+q}{16-25x^2}$, where&nbsp;$p$ and $q$ are constants, is $\frac{Ax+B}{25x^2-16}$.</p>
<p>Find an expression, in terms of $p$ and / or $q$, for&nbsp;$A$ and $B$.</p>

The sum of $\frac{3}{5x+4}$, $\frac{7}{4-5x}$ and $\frac{px+q}{16-25x^2}$, where $p$ and $q$ are constants, is $\frac{Ax+B}{25x^2-16}$.

Find an expression, in terms of $p$ and / or $q$, for $A$ and $B$.

Addition and Subtraction of Algebraic Fractions

Algebraic Expressions, Identities and Formulae

<p>Given that&nbsp;$3{{x}^{3}}+{{x}^{2}}-2x-3\equiv (Ax+B)(x-1)(x+2)+C(x-1)+D$, find the values of $A$, $B$, $C$ and $D$.<span class="inline-marks">[4]</span></p>

Given that $3{{x}^{3}}+{{x}^{2}}-2x-3\equiv (Ax+B)(x-1)(x+2)+C(x-1)+D$, find the values of $A$, $B$, $C$ and $D$.

IP3 Class Test (May 2025) Q2

<p>The coefficient of $x^3$ in the cubic polynomial, $\mathrm{f}(x)$, is $3$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the roots of the equation, $\mathrm{f}(x)=0$, are $-5$, $2p$ and $p$, find an expression for&nbsp;$\mathrm{f}(x)$ in terms of $p$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given also that the coefficient of $x$ is $-21$, find the possible value(s) of $p$.</span><span class="inline-marks">[3]</span></div></li></ol>

The coefficient of $x^3$ in the cubic polynomial, $\mathrm{f}(x)$, is $3$.

Given that the roots of the equation, $\mathrm{f}(x)=0$, are $-5$, $2p$ and $p$, find an expression for $\mathrm{f}(x)$ in terms of $p$.

Given also that the coefficient of $x$ is $-21$, find the possible value(s) of $p$.

Forming Polynomials from Roots and Factors

<p>If&nbsp;$2x^3-x^2-5x-4 =(Ax+3)(x-3)(x+1) +Bx+C$ for all real values of $x$ find $A$, $B$ and $C$.</p>

If $2x^3-x^2-5x-4 =(Ax+3)(x-3)(x+1) +Bx+C$ for all real values of $x$ find $A$, $B$ and $C$.

<p>A polynomial $\text{h}(x)=2{{x}^{3}}+A{{x}^{2}}+Bx-6$.  It is given that $2{{x}^{3}}+A{{x}^{2}}+Bx-6\equiv (x+3)(2{{x}^{2}}-3x-2)$. Find the value of $A$ and of $B$.</p>

A polynomial $\text{h}(x)=2{{x}^{3}}+A{{x}^{2}}+Bx-6$.  It is given that $2{{x}^{3}}+A{{x}^{2}}+Bx-6\equiv (x+3)(2{{x}^{2}}-3x-2)$. Find the value of $A$ and of $B$.

<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 $x$. Find the values of $A$, $B$ and $C$.</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 $x$. Find the values of $A$, $B$ and $C$.

<p>Given the identity $A{{x}^{3}}+{{x}^{2}}-2x-3\equiv (3x+B)(x-1)(x+2)+C(x-1)+D$, find the values of the constants $A$, $B$, $C$ and $D$.  </p>

Given the identity $A{{x}^{3}}+{{x}^{2}}-2x-3\equiv (3x+B)(x-1)(x+2)+C(x-1)+D$, find the values of the constants $A$, $B$, $C$ and $D$.  

<p>Find the values of the unknown constants $a$, $b$ and $c$ in $2{{x}^{3}}-3{{x}^{2}}-5x+18\equiv \left( x+2 \right)\left( a{{x}^{2}}+bx+c \right)$.</p>

Find the values of the unknown constants $a$, $b$ and $c$ in $2{{x}^{3}}-3{{x}^{2}}-5x+18\equiv \left( x+2 \right)\left( a{{x}^{2}}+bx+c \right)$.

Equating Coefficients

<p>If $2{{x}^{3}}-{{x}^{2}}-5x-4=(Ax+3)(x-3)(x+1)+Bx+C$ for all real values of $x$, find $A$, $B$ and $C$.</p>

If $2{{x}^{3}}-{{x}^{2}}-5x-4=(Ax+3)(x-3)(x+1)+Bx+C$ for all real values of $x$, find $A$, $B$ and $C$.

Finding Unknown Coefficients

<p><strong>Vieta's</strong> Formulae relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète. </p>
<p>For a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ with roots ${{x}_{1}}$ and ${{x}_{2}}$, it is satisfied that </p>
<p style="text-align: center">Sum of Roots: ${{x}_{1}}+{{x}_{2}}=-\frac{b}{a}$</p>
<p style="text-align: center">Product of Roots:  ${{x}_{1}}{{x}_{2}}=\frac{c}{a}$</p>
<p>Show that for a cubic equation in the form of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$, where&nbsp;$a\ne0$, with roots ${{x}_{1}}$, ${{x}_{2}}$ and ${{x}_{3}}$, Vieta’s Formulae are given by</p>
<p style="text-align: center">$\begin{matrix}
  {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-\frac{b}{a} \\ 
  {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{2}}{{x}_{3}}=\frac{c}{a} \\ 
  {{x}_{1}}{{x}_{2}}{{x}_{3}}=-\frac{\mathrm{d}}{a} \\ 
\end{matrix}$</p>

**Vieta's** Formulae relate the coefficients of a polynomial to sums and products of its roots. They are named after the French mathematician François Viète. 

For a quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ with roots ${{x}_{1}}$ and ${{x}_{2}}$, it is satisfied that 

Sum of Roots: ${{x}_{1}}+{{x}_{2}}=-\frac{b}{a}$

Product of Roots:  ${{x}_{1}}{{x}_{2}}=\frac{c}{a}$

Show that for a cubic equation in the form of $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$, where $a\ne0$, with roots ${{x}_{1}}$, ${{x}_{2}}$ and ${{x}_{3}}$, Vieta’s Formulae are given by

$\begin{matrix}
  {{x}_{1}}+{{x}_{2}}+{{x}_{3}}=-\frac{b}{a} \\ 
  {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+{{x}_{2}}{{x}_{3}}=\frac{c}{a} \\ 
  {{x}_{1}}{{x}_{2}}{{x}_{3}}=-\frac{\mathrm{d}}{a} \\ 
\end{matrix}$

IP3U2 HOT PN1 Q8

<p>Given that $\left( 3x-1 \right)\left( x+p \right)=3{{x}^{2}}+qx-5$, find the value of</p>
<ol class="abc-list"><li><div class="list-item-paragraph">$p$,</div></li><li><div class="list-item-paragraph">$q$.</div></li></ol>

Given that $\left( 3x-1 \right)\left( x+p \right)=3{{x}^{2}}+qx-5$, find the value of

IP2U3 Exercise 1 Q6

<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 $x$.

Polynomial Division and Remainders

<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 values of $A$, $B$ and $C$ given that $2{{x}^{3}}+5{{x}^{2}}-18x+14\equiv (Ax-1)(x-2)(x+5)+B(x+1)+C$.</p>

Find the values of $A$, $B$ and $C$ given that $2{{x}^{3}}+5{{x}^{2}}-18x+14\equiv (Ax-1)(x-2)(x+5)+B(x+1)+C$.

<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 $x$. Find the values of $A$, $B$ and $C$.</p>

Question 113764

<p>Given the identity $A{{x}^{3}}+{{x}^{2}}-2x-3\equiv (3x+B)(x-1)(x+2)+C(x-1)+D$, find the values of the constants $A$, $B$, $C$ and $D$.</p>

Given the identity $A{{x}^{3}}+{{x}^{2}}-2x-3\equiv (3x+B)(x-1)(x+2)+C(x-1)+D$, find the values of the constants $A$, $B$, $C$ and $D$.

Question 113756

<p>Given that $3{{x}^{3}}+{{x}^{2}}-2x-3=(Ax+B)(x-1)(x+2)+C(x-1)+D$ for all values of $x$, find the values of $A$, $B$, $C$ and $D$.</p>

Given that $3{{x}^{3}}+{{x}^{2}}-2x-3=(Ax+B)(x-1)(x+2)+C(x-1)+D$ for all values of $x$, find the values of $A$, $B$, $C$ and $D$.

<p>Given that $3{{x}^{3}}+4{{x}^{2}}-5x+2=(Ax+1)(x+2)(x+B)+C$ for all values of $x$, find the value of $A$, $B$ and $C$.</p>

Given that $3{{x}^{3}}+4{{x}^{2}}-5x+2=(Ax+1)(x+2)(x+B)+C$ for all values of $x$, find the value of $A$, $B$ and $C$.

Math Question Bank