Let $P(x)=x^3+ax^2+bx+c$. When $P(x)$ is divided by $x^2-3x+2$, the remainder is $5x-4$. When $P(x)$ is divided by $x-3$, the remainder is $11$.

A monic quartic polynomial has real coefficients, constant term $20$, and roots $1+2i$, $2$ and $2$.

The polynomial $p(x)=x^3+ax^2+bx+7$ leaves the same remainder when divided by each of $x-1$, $x-3$ and $x-5$.

Consider the polynomial

A quartic polynomial $P$ has real coefficients and positive leading coefficient. It has a double root at $x=1$, is divisible by $x^2+4$, and satisfies $P(0)=12$.

The polynomial $p(x)=x^4+kx^3-3x^2+lx+2$ has $(x-1)^2$ as a factor.

The roots of $p(x)=x^3-2x^2-5x+6$ are $\alpha$, $\beta$ and $\gamma$.

Let $p(x)$ be the monic cubic polynomial $p(x)=x^3+ax^2+bx+c$. The sum of the roots of $p(x)=0$ is $4$. It is also given that $x-2$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $x+1$ is $6$.

Consider the polynomial

The graph of $y=f(x)$, where

intersects a horizontal line $y=c$.

The cubic polynomial $p(x)=x^3-3x^2+kx+4$ has a repeated root.

The polynomial

has real coefficients. Four of its roots are $2+i$, $2-i$, $-1$ and $3$.

A monic quartic polynomial $P$ has real coefficients and is divisible by $x^2-2x+5$. The remainders when $P(x)$ is divided by $x-1$ and by $x+1$ are $12$ and $72$ respectively.

A polynomial function $p$ of degree $5$ has exactly two distinct zeros. The zero at $x=-2$ has multiplicity $2$, and the zero at $x=1$ has multiplicity $3$. It is also given that $p(0)=16$. Define $q(x)=p(2x-1)$.

A cubic polynomial $f(x)=ax^3+bx^2+cx+d$ satisfies

Let $p(x)=x^4-3x^3+ax^2+bx+5$. The remainder when $p(x)$ is divided by $x-1$ is $4$, and the remainder when $p(x)$ is divided by $x+2$ is $61$.

The equation $x^3-8x^2+kx-12=0$ has roots $\lpha$, $\beta$ and $\gamma$. It is given that $\gamma=\lpha+\beta$.

A monic quartic polynomial $p$ has real coefficients. Two of the roots of $p(x)=0$ are $1+2i$ and $2$. The sum of all four roots is $7$ and the product of all four roots is $30$.

For real $a$, define $p_a(x)=x^4+ax^3-6x^2+4x+8$.

The equation

has three positive real roots which are in arithmetic progression.

The monic cubic polynomial $p(x)=x^3+ax^2+bx-27$ has roots $\frac{3}{r}$, $3$ and $3r$, where $r\ne 0$. It is given that the remainder when $p(x)$ is divided by $x-1$ is $-2$.

Let $P(x)=x^4+ax^3+bx^2+cx+10$, where $a,b,c\in\mathbb{R}$. It is given that $x^2+1$ is a factor of $P(x)$ and that the remainder when $P(x)$ is divided by $x-1$ is $10$.

Let $p(x)=x^3-5x^2+2x+8$. The roots of $p(x)=0$ are denoted by $\alpha$, $\beta$ and $\gamma$. A new polynomial equation has roots $\frac{1}{\alpha-1}$, $\frac{1}{\beta-1}$ and $\frac{1}{\gamma-1}$.

Let

where $a,b,c\in\mathbb{R}$. It is given that $x^2-4$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $x+1$ is $30$.

The monic cubic polynomial $p(x)=x^3+ax^2+bx-80$ has three roots which are consecutive terms of an arithmetic sequence. It is given that the sum of the roots is $15$.

Let $p(x)=x^3-4x^2+x+6$. The roots of $p(x)=0$ are $\alpha$, $\beta$ and $\gamma$. A polynomial equation $q(x)=0$ has roots $\frac{1}{\alpha}$, $\frac{1}{\beta}$ and $\frac{1}{\gamma}$.

A monic quartic polynomial $P$ has real coefficients. Two of its roots are $2+i$ and $2-i$. The other two roots are equal. The coefficient of $x^3$ in $P(x)$ is $-8$.

Consider the polynomial $p(x)=x^5+ax^4+bx^3-7x^2+cx+10$, where $a$, $b$ and $c$ are real constants. It is given that $x^2-1$ is a factor of $p(x)$ and that the remainder when $p(x)$ is divided by $x-2$ is $12$.

The polynomial $p(x)=x^4-15x^3+70x^2-120x+64$ has roots $\alpha_1$, $\alpha_2$, $\alpha_3$ and $\alpha_4$. A new polynomial $q$ has roots $\beta_i=1+\dfrac{1}{\alpha_i}$, for $i=1,2,3,4$.

A cubic polynomial has leading coefficient $2$. When divided by $x-3$ and by $x+2$, the remainder is $0$ in each case. When divided by $x-1$, the remainder is $4$.

A symmetric bridge arch is modelled by the polynomial $h(x)=ax^4+bx^2+c$, where $h$ is the height in metres and $x$ is the horizontal distance in metres from the centre of the arch. The arch is $6$ metres high at its centre. At $x=4$, the arch meets the ground and has gradient $-5$.