The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x - 2)$ is 4 times the remainder when

$\displaystyle f(x)$ is divided by $\displaystyle (x + 1)$

(a) Show that $\displaystyle c = 6$

Given that $\displaystyle (x + 2)$ is a factor of $\displaystyle f(x)$

(b) show that the equation $\displaystyle f(x) = 0$ has only one real root.

Given that $\displaystyle (3x - 2)$ is a factor of $\displaystyle f(x)$

(a) show that $\displaystyle a = 21$

(b) Hence show algebraically that the curve $\displaystyle y = f(x)$ has only one intersection with the $\displaystyle x$-axis.

Figure 2 shows part of the curve with equation $\displaystyle y = \frac{x^2}{3} - \frac{1}{2x}$ for $\displaystyle -4 < x < 0$

By drawing a suitable straight line on the grid, obtain estimates, to one decimal place,

of the roots of the equation $\displaystyle 4x^3 + 3x^2 - 36x - 6 = 0$ in the interval $\displaystyle -4 < x < 0$

$\displaystyle f(x) = px^3 + qx^2 - 37x - 12q$ where $\displaystyle p$ and $\displaystyle q$ are constants.

When $\displaystyle f'(x)$ is divided by $\displaystyle (x + 2)$ the remainder is $\displaystyle -33$

Given that $\displaystyle (x + 5)$ is a factor of $\displaystyle f(x)$

(a) (i) show that $\displaystyle p = 2$

(ii) find the value of $\displaystyle q$

(b) Hence, use algebra to factorise $\displaystyle f(x)$ completely.

(c) Hence solve the equation $\displaystyle f(x) = 0$

Figure 1 shows part of the curve with equation $\displaystyle y = \frac{x}{2} + \frac{4}{x^2}$ in the interval $\displaystyle 0.8 < x < 7$

By drawing a suitable straight line on the grid, obtain an estimate, to one decimal place, of the roots of the equation $\displaystyle 3x^3 - 12x^2 + 8 = 0$ in the interval $\displaystyle 0.8 < x < 7$

Given that $\displaystyle (2x - 1)$ is a factor of $\displaystyle f(x)$

show that the curve with equation $\displaystyle y = f(x)$ has only one intersection with the $\displaystyle x$-axis.

The curve $\displaystyle y = g(x)$ passes through the point with coordinates $\displaystyle (1, 20)$

Given that $\displaystyle (x-5)$ is a factor of $\displaystyle g(x)$

(a) show that $\displaystyle g(x) = 2x^3 - 5x^2 - 37x + 60$

(b) Hence, or otherwise, use algebra to solve the equation $\displaystyle g(x) = 0$

where $\displaystyle p$ and $\displaystyle q$ are integers. $\displaystyle (x+1)$ is a factor of $\displaystyle f(x)$. The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x+2)$ is $\displaystyle -5$.

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(b) Hence show that $\displaystyle f(x)=0$ has only one real root.

where $\displaystyle p$ and $\displaystyle q$ are integers.

Given that $\displaystyle (x-3)$ is a factor of $\displaystyle f(x)$,

(a) show that

Given that $\displaystyle (x+p)$, where $\displaystyle p>0$, is also a factor of $\displaystyle f(x)$,

(b) show that

(c) Hence find the value of $\displaystyle p$ and the value of $\displaystyle q$.

(d) Using your values of $\displaystyle p$ and $\displaystyle q$, factorise $\displaystyle f(x)$ completely.

where $\displaystyle p$ and $\displaystyle q$ are constants. The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x-1)$ is $\displaystyle -12$. The remainder when $\displaystyle f(x)$ is divided by $\displaystyle (x-4)$ is $\displaystyle 30$.

(a) Find the value of $\displaystyle p$ and the value of $\displaystyle q$.

Using your values of $\displaystyle p$ and $\displaystyle q$,

(b) show that $\displaystyle f(3)=0$,

(c) express $\displaystyle f(x)$ as a product of linear factors,

(d) hence solve $\displaystyle f(x)=0$.

The function $\displaystyle f$ is defined by

(a) Factorise

(b) Hence, or otherwise, show that $\displaystyle (x^2-x-2)$ is a factor of $\displaystyle f(x)$.

(a) Show that

(b) Hence use algebra to solve the equation

Give your roots to 3 decimal places where appropriate.

A curve $\displaystyle C$ has equation

(a) Write down an equation of the asymptote to $\displaystyle C$ that is

(i) parallel to the $\displaystyle y$-axis,

(ii) parallel to the $\displaystyle x$-axis.

(b) Find the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

(c) Using calculus show that at every point on the curve, the gradient of $\displaystyle C$ is positive.

(d) Using the axes provided, sketch $\displaystyle C$, showing clearly the asymptotes and the coordinates of the points of intersection of $\displaystyle C$ with the coordinate axes.

The line $\displaystyle l$ is the tangent to $\displaystyle C$ at the point on the curve where $\displaystyle x=1$.

(e) Find an equation of $\displaystyle l$, giving your answer in the form

Given that $\displaystyle f(0)=-12$,

(a) find the value of $\displaystyle q$.

(b) Find the range of values of $\displaystyle p$ for which the cubic equation $\displaystyle f(x)=0$ has only one real root.

Figure 1 shows part of the curve $\displaystyle S$ with equation

where $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are integers.

The asymptote to $\displaystyle S$ parallel to the $\displaystyle x$-axis is $\displaystyle y=-2$.

The asymptote to $\displaystyle S$ parallel to the $\displaystyle y$-axis is $\displaystyle x=-3$.

The curve crosses the $\displaystyle x$-axis at $\displaystyle (4,0)$.

The curve crosses the $\displaystyle y$-axis at $\displaystyle (0,p)$, where $\displaystyle p$ is rational.

Find

(i) $\displaystyle a$,

(ii) $\displaystyle b$,

(iii) $\displaystyle c$,

(iv) $\displaystyle p$.