Polynomials (OCR A Level Maths A: Pure): 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.

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.

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

A cuboid has a length of $(2x-3y+3)$ units, a width of $(2x+3y-3)$ units, and a height of $(x-y)$ units.  Find an expression for the volume of the cuboid in terms of $x$ and $y$.

Given that $(ax+by)(2x+y)(x-3y)=8x^3+c{x}^{2}y+dx{y}^2-9y^3$, where $a,b,c$ and $d$ are constants, find the values of $a,b,c$ and $d$.

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

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