Polynomials (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

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

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

Divide $x^3-6x^2-9x+14$ by $\left(x-7\right)$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Show that $\left(2x-3\right)$ is a factor of $2x^3-13x^2+23x-12$.

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

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

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

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

(i) Write down the divisor.

(ii) Write down the quotient.

(iii) Write down the remainder.

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

(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

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

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

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

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

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

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

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

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

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

Show that $\left(x+2\right)$ is a factor of $2x^3-3x^2-11x+6$.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Divide $4x^4-37x^2+9$ by $\left(2x-1\right)$.

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

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

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

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

It is given that

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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