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

Expand and simplify $y(2x+2)(7-x)$.

A rectangle has side lengths of $(3x-2y+5)$ units and $(x+3y-1)$ units. Find an expression for the area of the rectangle in terms of $x$ and $y$.

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

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

Divide $x^3-6x^2-9x+14$ by $(x-7)$.

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{f}\left(x\right)=x^3+9x^2+rx+s$. Given that $\mathrm{f}\left(2\right)=0$and $\mathrm{f}(-1)=-54$:

find the values of $r$ and $s$.

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

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

$\mathrm{f}\left(x\right)=3x^3-5x^2-4x+4$

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

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

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

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

Show that$(2x-3)$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 $(2x-1)$ 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

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

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

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

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

Consider the function $f\left(x\right)=4x^3+6x^2-7x+2.$

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

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

The function $f\left(x\right)=2x^3-5x^2+ax+b$  has $(2x+3)$ as a factor, and when $f\left(x\right)$ is divided by $(x-2)$ the remainder is 7. 

Show that $a$ and $b$ must satisfy the simultaneous equations:  

$2a+b=11$

$3a-2b=-36$

Hence find $a$ and $b.$

Show that $(x+2)$ 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.

Expand and simplify $(2-x)(3x+1)(x+1)$.

A square has side lengths of $(5x-2y+3)$ units. Find an expression for the length of the diagonal of the square in terms of $x$ and $y$.

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

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

Divide $x^3-19x-30$ by $(x-5)$.

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{f}\left(x\right)=x^3+r{x}^2+sx-30$. Given that $\mathrm{f}\left(2\right)=0$ and $\mathrm{f}(-3)=-240$: 

find the values of $r$ and $s$.

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

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

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

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

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

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

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

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

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

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

(i) Find the remainder when  $x^3-2x^2+4x-3$  is divided by $x-2.$

  [2]

(ii) Find the value of $\mathrm{f}\left(2\right)$ when $\mathrm{f}\left(x\right)=x^3-2x^2+4x-3$.

  [1]

(iii) Comment on your answers to parts (i) and (ii).

[1]

It is given that

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

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

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

Explain, with an example, why the forms of $\mathrm{f}\left(x\right)$and $\mathrm{g}\left(x\right)$ 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  $x+2$ the quotient is $x^2-4x+12$ and the remainder is $b$.

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

Given that $\left(x+4\right)$ is a factor of the function $f\left(x\right)=p{x}^3+(5p+1)x^2+5qx-2q-2$ and that the remainder when $f\left(x\right)$  is divided by $\left(x+1\right)$ is $-12$,  find the values of the constants p and q.

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

Given that $(x+1)$ 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 $(x+3)$ is a factor of the function $\mathrm{f}\left(x\right)=6x^3+23x^2+11x-12$ and hence, or otherwise, fully factorise $\mathrm{f}\left(x\right)$.

Sketch the graph of $y=\mathrm{f}\left(x\right)$.

Label any points where the graph crosses the coordinate axes.

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.

Expand and simplify $(x+y)(x-y)(y-x)(-x-y)$.