Polynomials (Edexcel International AS Maths: Pure 2): Exam Questions

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 $a,b$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.

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 $b.$

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.

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.

Divide $4x^4-37x^2+9$ by $(2x-1)$.

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

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

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

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

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

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

Hence show that the equation $\mathrm{f}\left(x\right)=0$ has exactly 2 real roots.

 Given that 3 is a root of the equation $2x^3-x^2-11x-12=0$, prove that the equation has no other real roots.

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

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

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