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<p>Suppose $\mathrm P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $\mathrm P(1) = 1, \mathrm P(2) = 2, \mathrm P(3) = 3, \mathrm P(4) = 4$, it is also given&nbsp;$\mathrm P(5) = -19$.<br>Find $\mathrm P(6)$.</p>

Suppose $\mathrm P(x) = ax^4 + bx^3 + cx^2 + dx + e$ satisfies $\mathrm P(1) = 1, \mathrm P(2) = 2, \mathrm P(3) = 3, \mathrm P(4) = 4$, it is also given $\mathrm P(5) = -19$.
Find $\mathrm P(6)$.

Forming Polynomials from Roots and Factors

Polynomials, Cubic Equations and Partial Fractions

<p>A cubic polynomial satisfies $\mathrm P(1) = \mathrm P(2) =\mathrm P(3) = 7$. Also $ \mathrm P(0) = 1$<br>Find $ \mathrm P(4)$</p>


A cubic polynomial satisfies $\mathrm P(1) = \mathrm P(2) =\mathrm P(3) = 7$. Also $ \mathrm P(0) = 1$
Find $ \mathrm P(4)$

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<p>It is given that&nbsp;$\mathrm{f}(x)$ is a cubic function. </p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $\mathrm{f}(x)$ in the factorised form.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Hence find the value of $\mathrm{f}(1)$.<span class="inline-marks">[1]</span></div></li></ol>

It is given that $\mathrm{f}(x)$ is a cubic function. 

Polynomial Functions and Key Features

Functions and Graphs

<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

Remainder and Factor Theorems

IP3 Class Test (May 2025) Q6

<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 $p$.

Comparing Coefficients in Polynomial Identities

<p>Given that a cubic polynomial $\mathrm{f}\left( x \right)$ has a factor $x$ and $\mathrm{f}\left( x+1 \right)-\mathrm{f}\left( x \right)=3{{x}^{2}}+7x$,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">by substituting suitable values of $x$,</div><ol class="i-list"><li><div class="list-item-paragraph">show that&nbsp;$\mathrm{f}\left( x \right)$ has a factor of $x-1$,</div></li><li><div class="list-item-paragraph">show that&nbsp;$\mathrm{f}\left( x \right)$ has a remainder of $10$ when it is divided by $x-2$,</div></li><li><div class="list-item-paragraph">find the remainder when&nbsp;$\mathrm{f}\left( x \right)$ is divided by $x+1$.</div></li></ol></li><li><div class="list-item-paragraph">Find $\mathrm{f}\left( x \right)$</div></li></ol>

Given that a cubic polynomial $\mathrm{f}\left( x \right)$ has a factor $x$ and $\mathrm{f}\left( x+1 \right)-\mathrm{f}\left( x \right)=3{{x}^{2}}+7x$,

Algebraic Expressions, Identities and Formulae

<p>The term containing the highest power of $x$ in the polynomial $\mathrm{f}(x)$ is $5{{x}^{3}}$. The roots of the equation $\mathrm{f}(x)=0$ are $-2$, $\mathrm{2}$ and $p$.</p>
<p>Find,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">an expression, in terms of $x$ and $p$, for $\mathrm{f}(x)$ in descending powers of $x$,</div></li><li><div class="list-item-paragraph">the value of $p$ for which the division of $\mathrm{f}(x)$ by $x+1$ gives a remainder of $-30$,</div></li><li><div class="list-item-paragraph">the remainder when $\mathrm{f}(x)$ is divided by $2x+1$.</div></li></ol>

The term containing the highest power of $x$ in the polynomial $\mathrm{f}(x)$ is $5{{x}^{3}}$. The roots of the equation $\mathrm{f}(x)=0$ are $-2$, $\mathrm{2}$ and $p$.

Finding an Unknown Polynomial

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

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

<p>The term containing the highest power of $x$ in the polynomial $\mathrm{f}(x)$ is $5{{x}^{3}}$. The roots of the equation $\mathrm{f}(x)=0$ are $-2$, $\mathrm{2}$ and $p$.</p>
<p>Find,</p>
<ol class="abc-list"><li><div class="list-item-paragraph">an expression, in terms of $x$ and $p$, for $\mathrm{f}(x)$ in descending powers of $x$,</div></li><li><div class="list-item-paragraph">the value of $p$ for which the division of $\mathrm{f}(x)$ by $x+1$ gives a remainder of $-30$,</div></li><li><div class="list-item-paragraph">the remainder when $\mathrm{f}(x)$ is divided by $2x+1$.</div></li></ol>

Question 113986

<p>The cubic polynomial $\mathrm{h}\left( x \right)$ is such that the coefficient of ${{x}^{3}}$ is $4$ and the roots of the equation $\mathrm{h}\left( x \right)=0$ are $-2$, $5$, and $k$. Given that&nbsp;$\mathrm{h}\left( x \right)$ has a remainder of $-48$ when divided by $x-2$, show that&nbsp;$k=1$ and find the remainder when&nbsp;$\mathrm{h}\left( x \right)$ is divided by $4x+20$.</p>

The cubic polynomial $\mathrm{h}\left( x \right)$ is such that the coefficient of ${{x}^{3}}$ is $4$ and the roots of the equation $\mathrm{h}\left( x \right)=0$ are $-2$, $5$, and $k$. Given that $\mathrm{h}\left( x \right)$ has a remainder of $-48$ when divided by $x-2$, show that $k=1$ and find the remainder when $\mathrm{h}\left( x \right)$ is divided by $4x+20$.

<p>The cubic polynomial $\mathrm{g}\left( x \right)$ is such that the coefficient of ${{x}^{3}}$ is $3$ and the roots of the equation $\mathrm{g}\left( x \right)=0$ are $1$, $2$, and $k$. Given that&nbsp;$\mathrm{g}\left( x \right)$ has a remainder of $10$ when divided by $x-3$, show that&nbsp;$k=\frac{4}{3}$. Hence or otherwise, find the remainder when&nbsp;$\mathrm{g}\left( x \right)$ is divided by $3x-6$.</p>

The cubic polynomial $\mathrm{g}\left( x \right)$ is such that the coefficient of ${{x}^{3}}$ is $3$ and the roots of the equation $\mathrm{g}\left( x \right)=0$ are $1$, $2$, and $k$. Given that $\mathrm{g}\left( x \right)$ has a remainder of $10$ when divided by $x-3$, show that $k=\frac{4}{3}$. Hence or otherwise, find the remainder when $\mathrm{g}\left( x \right)$ is divided by $3x-6$.

<p>The term containing the highest power of $x$ in the polynomial $\mathrm{g}\left( x \right)$ is $4{{x}^{3}}$.</p>
<p>The roots of the equation $\mathrm{g}\left( x \right)=0$ are $-3$, $1$, and $q$.</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-paragraph">an expression, in terms of $x$ and $q$, for&nbsp;$\mathrm{g}\left( x \right)$ in descending powers of $x$,</div></li><li><div class="list-item-paragraph">the value of $q$ for which the division of $\mathrm{g}\left( x \right)$ by $x+2$ gives a remainder of $-24$,</div></li><li><div class="list-item-paragraph">the remainder when&nbsp;$\mathrm{g}\left( x \right)$ is divided by $3x-1$.</div></li></ol>

The term containing the highest power of $x$ in the polynomial $\mathrm{g}\left( x \right)$ is $4{{x}^{3}}$.

The roots of the equation $\mathrm{g}\left( x \right)=0$ are $-3$, $1$, and $q$.

an expression, in terms of $x$ and $q$, for $\mathrm{g}\left( x \right)$ in descending powers of $x$,

the value of $q$ for which the division of $\mathrm{g}\left( x \right)$ by $x+2$ gives a remainder of $-24$,

the remainder when $\mathrm{g}\left( x \right)$ is divided by $3x-1$.

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