Polynomials (Cambridge International A-Level Mathematics): Exam Questions

Use the factor theorem to verify that $( x - 2 )$ is a factor of $x^{3} - x^{2} - 14 x + 24$.

Factorise

(i) $4 x^{2} - 4 x - 15$

(ii) $3 x^{3} + 11 x^{2} - 4 x$

Divide $x^{3} + 6 x^{2} + 11 x + 6$ by $( x + 2 )$.

Given $\text{f} ( x ) = x^{3} - 5 x^{2} - 2 x + 24$ has a root at $x = 3$, fully factorise $\text{f} ( x )$.

Use polynomial division to show that $( x - 2 )$ is a factor of $2 x^{3} + 3 x^{2} - 18 x + 8$.

Given $x = - 2$ is a root of the function $\text{f} ( x ) = x^{3} - 3 x^{2} - 8 x + 4$, fully factorise $\text{f} ( x )$.

Given that $( x - 4 )$ is a factor of $x^{3} - k x^{2} - 4 x + 16$, find the value of $k$.

Given that $x = \frac{1}{2}$ is a root of the function $\text{f} ( x ) = 2 x^{3} + ( p^{2} + 1 ) x^{2} - 11 x + 4$, find the possible values of $p$.

Given that $( x + 1 )$ is a factor of $\text{f} ( x ) = x^{3} - 5 x^{2} + 3 x + 9$, fully factorise $\text{f} ( x )$.

Sketch the graph of $y = \text{f} ( x )$, labelling the coordinates of all points where the graph intersects the coordinate axes.

Find the remainder when $x^{3} - 7 x - 6$ is divided by $( x + 4 )$.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = x^{4} + a x^{3} - 13 x^{2} - 38 x - 4 b$, where $a$ and $b$ are constants.

Given that $( x + 1 )$ and $( x + 2 )$ are factors of $\text{f} ( x )$, find the values of $a$ and $b$.

The function $\text{f} ( x )$ is given by $\text{f} ( x ) = 2 x^{3} + 7 x^{2} - 4 x$. Show that $\text{f} ( x ) = x ( 2 x - 1 ) ( x + 4 )$.

Hence, or otherwise, write down the real solutions to the equation $\frac{\text{f}(x)}{x+1} = 0$.

The function $\text{f} ( x )$ is given by $\text{f} ( x ) = x^{3} - 4 x^{2} - 7 x + 10$. Work out $\text{f} ( 1 )$ and hence write down a factor of $\text{f} ( x )$.

Work out $\text{f} ( x ) \div ( x + 2 )$.

Write $\text{f} ( x )$ in the form $( x + a ) ( x + b ) ( x + c )$ where $a , b$ and $c$ are integers to be found.

Find the remainder when $x^{3} + 2 x^{2} - 5 x + 8$ is divided by $( x - 3 )$.

Given that $( x^{2} - 8 x - 20 ) \div ( x - 2 ) = A x + B + \frac{C}{x-2}$, where $A , B$ and $C$ are integer constants, in terms of $A , B$ and/or $C$ as appropriate:

(i) write down the divisor,

(ii) write down the quotient,

(iii) write down the remainder.

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

The function $\text{f}$ is defined by $\text{f} ( x ) = x^{2} + a x + b$ for $x \in ℝ$, where $a$ and $b$ are constants.

Given that $\text{f} ( 3 ) = 0$ and $\text{f} ( - 8 ) = 0$, find the values of $a$ and $b$.

Use the factor theorem to show that $( x - 2 )$ is a factor of the function $\text{f} ( x ) = x^{3} - 2 x^{2} - 4 x + 8$.

Hence, or otherwise, express $\text{f} ( x )$ as a product of three linear factors.

Sketch the graph of $y = \text{f} ( x )$, labelling any points where the graph intersects the coordinate axes.

Divide $x^{3} - 19 x - 30$ by $( x - 5 )$.

For a polynomial $\text{f} ( x )$, the Remainder Theorem states that when $\text{f} ( x )$ is divided by $( a x - b )$ the remainder is \text{f}\left(\dfrac{b}{a}\right).

Use the Remainder Theorem to find the remainder when $8 x^{3} + 6 x^{2} - x - 2$ is divided by $( 2 x + 1 )$.

Work out the remainder when $6 x^{2} - x - 2$ is divided by $( 2 x + 1 )$.

Given that $( 3 x + y ) ( 2 x - 3 y ) ( x - 2 y ) = a x^{3} + b x^{2} y + c x y^{2} + d y^{3}$, where $a , b , c$ and $d$ are constants, find the values of $a , b , c$ and $d$.

Factorise completely $3 x^{3} - 51 x^{2} + 126 x$.

Show that $( x + 2 )$ is a factor of $2 x^{3} - 3 x^{2} - 11 x + 6$.

Fully factorise $2 x^{3} - 3 x^{2} - 11 x + 6$.

Sketch the graph of $y = 2 x^{3} - 3 x^{2} - 11 x + 6$. Label any points where the graph crosses the coordinate axes.

Divide $x^{3} - 6 x^{2} - 9 x + 14$ by $( x - 7 )$.

$\text{f} ( x ) = 2 x^{3} - x^{2} - 16 x + 15$

Find the remainder when $\text{f} ( x )$ is divided by $( x - 2 )$.

Given that $( x + 3 )$ is a factor of $\text{f} ( x )$, factorise $\text{f} ( x )$ completely.

$\text{f} ( x ) = 2 x^{3} - 3 x^{2} - 72 x - 35$

Show that $\text{f} ( x ) = ( 2 x + 1 ) ( a x^{2} + b x + c )$ where $a , b$ and $c$ are constants to be found.

Hence factorise $\text{f} ( x )$ completely.

Write down all the real roots of the equation $\text{f} ( x ) = 0$.

$\text{f} ( x ) = 4 x^{3} + 4 x^{2} - 23 x - 30$

Use the factor theorem to show that $( x + 2 )$ is a factor of $\text{f} ( x )$.

Factorise $\text{f} ( x )$ completely.

Write down all the real roots of the equation $\text{f} ( x ) = 0$.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = x^{3} + 9 x^{2} + r x + s$.

Given that $\text{f} ( 2 ) = 0$ and $\text{f} ( - 1 ) = - 54$, find the values of $r$ and $s$.

Factorise $\text{f} ( x )$ completely.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = 3 x^{3} - 5 x^{2} - 4 x + 4$.

Show that \text{f}\left(\frac{2}{3}\right) = 0.

Hence write down a factor of $\text{f} ( x )$.

Fully factorise $\text{f} ( x )$.

Write down the solutions to the equation $\text{f} ( x ) = 0$.

Show that $( 2 x - 3 )$ is a factor of $2 x^{3} - 13 x^{2} + 23 x - 12$.

Fully factorise $2 x^{3} - 13 x^{2} + 23 x - 12$.

Find all the real solutions to $2 x^{3} - 13 x^{2} + 23 x - 12 = 0$.

Given that $( 2 x - 1 )$ is a factor of $2 x^{3} + x^{2} - 25 x + a$, find the value of $a$.

Given $( x^{2} + 8 x - 4 ) \div ( x - 3 ) = x + 11 + \frac{29}{x-3}$

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

(i) Write down the degree of $x^{2} + 8 x - 4$.

(ii) Write down the degree of $x - 3$.

(iii) Explain why you would expect the quotient to be of degree 1 in this case.

It is given that

$\frac{\text{f}(x)}{x+2} = 3 x + 4 - \frac{2}{x+2}$

Find $\text{f} ( x )$.

The result of dividing $x^{2} + a x - 5$ by $( x + 1 )$ is $x + 3 + \frac{d}{x+1}$. Find the values of $a$ and $d$.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = 4 x^{3} + 6 x^{2} - 7 x + 2$.

(i) Find the quotient and the remainder when $4 x^{3} + 6 x^{2} - 7 x + 2$ is divided by $( x - 2 )$.

(ii) Hence express $4 x^{3} + 6 x^{2} - 7 x + 2$ in the form $( x - 2 ) ( a x^{2} + b x + c ) + d$, where $a , b , c$ and $d$ are constants to be determined.

The polynomial $\text{f} ( x )$ is defined by $\text{f} ( x ) = 2 x^{3} - 5 x^{2} + a x + b$, where $a$ and $b$ are constants. It is given that $( 2 x + 3 )$ is a factor of $\text{f} ( x )$, and that when $\text{f} ( x )$ is divided by $( x - 2 )$ the remainder is $7$.

Show that $a$ and $b$ satisfy the equations $2 a + b = 11$ and $3 a - 2 b = - 36$.

Hence find the values of $a$ and $b$.

Given that $( x + 1 )$ is a factor of $x^{3} - 4 x^{2} + x + 6$, sketch the graph of $y = x^{3} - 4 x^{2} + x + 6$. Label any points where the graph intersects the coordinate axes.

(There is no need to label any stationary points.)

Show that $( x + 3 )$ is a factor of the function $\text{f} ( x ) = 6 x^{3} + 23 x^{2} + 11 x - 12$ and hence, or otherwise, fully factorise $\text{f} ( x )$.

Sketch the graph of $y = \text{f} ( x )$. Label any points where the graph crosses the coordinate axes.

Factorise completely $15 x^{3} + 19 x^{2} - 10 x$.

$\text{f} ( x ) = x^{3} - 28 x + 48$

Find the remainder when $\text{f} ( x )$ is divided by $( x - 3 )$.

Given that $( x + 6 )$ is a factor of $\text{f} ( x )$, factorise $\text{f} ( x )$ completely.

Show that $( 5 x - 2 )$ is a factor of $25 x^{3} + 55 x^{2} - 56 x + 12$.

Hence find all the real solutions to the equation $25 x^{3} + 55 x^{2} - 56 x + 12 = 0$.

$\text{f} ( x ) = x^{3} + r x^{2} + s x - 30$

Given that $\text{f} ( 2 ) = 0$ and $\text{f} ( - 3 ) = - 240$, find the values of $r$ and $s$.

Factorise $\text{f} ( x )$ completely.

$\text{f} ( x ) = 4 x^{3} - 7 x - 3$

Use the factor theorem to show that $( 2 x + 1 )$ is a factor of $\text{f} ( x )$.

Factorise $\text{f} ( x )$ completely.

Write down all the real roots of the equation $\text{f} ( x ) = 0$.

$\text{f} ( x ) = 6 x^{3} - 19 x^{2} + 11 x + 6$

Show that $\text{f} ( x ) = ( 2 x - 3 ) ( a x^{2} + b x + c )$ where $a , b$ and $c$ are constants to be found.

Hence factorise $\text{f} ( x )$ completely.

Write down all the real roots of the equation $\text{f} ( x ) = 0$.

Given that $( 4 x - 5 )$ is a factor of $4 x^{3} - 9 x^{2} + a x + 30$, find the value of $a$.

Hence, or otherwise, fully factorise $4 x^{3} - 9 x^{2} + a x + 30$.

The function $\text{f} ( x )$ is given by

$\text{f} ( x ) = 4 x^{3} - 7 x^{2} - 21 x + 18$

Show that $( 4 x - 3 )$ is a factor of $\text{f} ( x )$.

Hence, or otherwise, fully factorise $\text{f} ( x )$.

Write down the roots of $\text{f} ( x )$.

Show that $3 x^{3} + 16 x^{2} - 22 x$ can be written in the form $( 3 x + 1 ) ( a x^{2} + b x + c ) + d$, where $a , b , c$ and $d$ are constants to be found.

Divide $4 x^{4} - 37 x^{2} + 9$ by $( 2 x - 1 )$.

Given that $( 2 x - 5 )$ is a factor of the function

$\text{f} ( x ) = 2 x^{3} + k x^{2} - 11 x - 60$

find the value of $k$ and fully factorise $\text{f} ( x )$.

Find the coordinates of the points of intersection between the curve with equation $y = x^{3} - x^{2} - 4 x + 4$ and the line with equation $y = 2 x + 4$.

On the same diagram, sketch the graphs of $y = x^{3} - x^{2} - 4 x + 4$ and $y = 2 x + 4$.

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

It is given that

$\frac{\text{f}(x)}{\text{g}(x)} = 2 x + 3 - \frac{4}{x+1}$

Why would assuming that $\text{g} ( x ) = x + 1$ be a logical first step in attempting to determine the precise forms of $\text{f} ( x )$ and $\text{g} ( x )$?

By first making the assumption from part (a), find $\text{f} ( x )$.

Explain, with an example, why the forms of $\text{f} ( x )$ and $\text{g} ( x )$ determined in parts (a) and (b) are not the only possible forms for those functions.

When $x^{3} + a x^{2} + 4 x - 1$ is divided by $( x + 2 )$ the quotient is $x^{2} - 4 x + 12$ and the remainder is $b$.

Find the values of $a$ and $b$.

The polynomial $p x^{3} + ( 5 p + 1 ) x^{2} + 5 q x - 2 q - 2$, where $p$ and $q$ are constants, is denoted by $\text{f} ( x )$.

It is given that $( x + 4 )$ is a factor of $\text{f} ( x )$, and that when $\text{f} ( x )$ is divided by $( x + 1 )$ the remainder is $- 12$.

Find the values of $p$ and $q$.

On the same diagram, sketch the graphs of $4 y = x^{3} - 5 x^{2} - 12 x + 36$ and $x + y - 6 = 0$.

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

$\text{f} ( x ) = 6 x^{4} + 7 x^{3} - 27 x^{2} - 28 x + 12$

Find the remainder when $\text{f} ( x )$ is divided by $( 2 x + 3 )$.

Given that $( x + 2 )$ is a factor of $\text{f} ( x )$, factorise $\text{f} ( x )$ completely.

$\text{f} ( x ) = 2 x^{4} - 15 x^{3} - 10 x^{2} + 105 x + 98$

Show that $\text{f} ( - 1 ) = 0$ and $\text{f} ( - 2 ) = 0$.

Hence, solve $\text{f} ( x ) = 0$.

When $2 x^{3} + ( a + b ) x^{2} + ( a - b ) x - 3$ is divided by $x + 4$ the quotient is $2 x^{2} + ( 2 a + 3 ) x + ( 2 b - 5 )$ and the remainder is $c$.

Find the values of $a$, $b$ and $c$.

$( x - 2 )$ and $( x + 3 )$ are factors of $\text{f} ( x )$, where $\text{f} ( x ) = x^{4} - 9 x^{3} + 9 x^{2} + 85 x - 150$.

Sketch the graph of $y = \text{f} ( x )$, labelling any points where the graph intersects the coordinate axes. (There is no need to label any stationary points.)

Sketch the graph of $y = 3 x^{3} + 2 x^{2} - 3 x + 10$, labelling any points where the graph intersects the coordinate axes.

Fully factorise $4 x^{3} + 17 x^{2} + 20 x + 4$.

Sketch the graph of $y = 4 x^{3} + 17 x^{2} + 20 x + 4$. Label any points where the graph crosses the coordinate axes.

On the same diagram, sketch the graphs of $y = x^{3} - 3 x^{2} - 6 x + 8$ and $y = \frac{3}{x^{2}}$.

Label any points where the graphs intersect the coordinate axes.

Write down the number of solutions to the equation

$x^{5} - 3 x^{4} - 6 x^{3} + 8 x^{2} = 3$.

Factorise completely $x^{5} y - x y^{5}$.

$\text{f} ( x ) = 3 x^{4} + x^{3} - 12 x^{2} - 49 x - 15$

Show that $\text{f} ( x ) = ( 3 x + 1 ) ( a x^{3} + b x^{2} + c x + d )$ where $a , b , c$ and $d$ are constants to be found.

Given that $( x - 3 )$ is a factor of $\text{f} ( x )$, factorise $\text{f} ( x )$ completely.