Polynomials (AQA A Level Maths: Pure): Exam Questions

A function is given by 

$\mathrm{f}\left(x\right)=x^3-5x^2-2x+24$

The equation $\mathrm{f}\left(x\right)=0$ has a solution at $x=3$.

Use algebra to factorise $\mathrm{f}\left(x\right)$ into three linear factors.

Show that

$(3x+y)(2x-3y)(x-2y)\equiv a{x}^3+b{x}^{2}y+cx{y}^2+d{y}^3$

where $a$, $b$, $c$ and $d$ are constants to be found.

Use polynomial division to divide $x^3-19x-30$ by $(x-5)$.

A function is defined by

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

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

A function is defined by

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

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 $\mathrm{f}\left(x\right)$ into three linear factors.

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

$\mathrm{f}\left(x\right)=2x^3+5x^2+2x+15$

Use the factor theorem to show that $\left(x+3\right)$ is a factor of $\mathrm{f}\left(x\right)$.

Find the constants $a$, $b$ and $c$ such that

$\mathrm{f}\left(x\right)=\left(x+3\right)\left(a{x}^2+bx+c\right)$

Hence show that $\mathrm{f}\left(x\right)=0$ has only one real root.

Write down the real root of the equation $\mathrm{f}\left(x-5\right)=0$

A function is given by

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

Show that $(x+2)$is a factor of $\mathrm{f}\left(x\right)$.

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

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

$\mathrm{f}\left(x\right)=(x-4)(x^2-3x+k)-42$ where $k$ is a constant

Given that $(x+2)$ is a factor of $\mathrm{f}\left(x\right)$, find the value of $k$.

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

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

A function is given by

$\mathrm{f}\left(x\right)=x^3-5x^2+3x+9$

Given that $(x+1)$ is a factor, use an algebraic method to factorise $\mathrm{f}\left(x\right)$.

Give your answer in the form

$\mathrm{f}\left(x\right)=\left(x+p\right){\left(x+q\right)}^2$

where $p$ and $q$ are integers to be found.

Sketch the curve with equation $y=\mathrm{f}\left(x\right)$, labelling the coordinates of any points at which the curve meets the coordinate axes.

Show that

$(2x-3y{)}^2(y-2x)=a{x}^3+b{x}^{2}y+cx{y}^2+d{y}^3$

where $a$, $b$, $c$ and $d$ are constants to be found.

A function is given by

$\mathrm{f}\left(x\right)=2x^3+(p^2+1)x^2-11x+4$

Given that $x=\frac{1}{2}$ is a root of the equation $\mathrm{f}\left(x\right)=0$, find the possible values of $p$.

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

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.

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

A function is given by

$\mathrm{f}\left(x\right)=x^3-3x^2-8x+4$

Given that $x=-2$ is a solution to the equation $\mathrm{f}\left(x\right)=0$, use algebra to factorise $\mathrm{f}\left(x\right)$ as far as possible.

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

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.

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$\mathrm{f}\left(x\right)=4x^3+5x^2-10x+4ax\in \mathrm{ℝ}$

where $a$ is a positive constant.

Given $\left(x-a\right)$ is a factor of $\mathrm{f}\left(x\right)$, show that

$a\left(4a^2+5a-6\right)=0$

Hence

(i) Find the value of $a$

(ii) use algebra to find the exact solutions of the equation

$\mathrm{f}\left(x\right)=3$

$\mathrm{f}\left(x\right)=-3x^3+8x^2-9x+10,x\in \mathrm{ℝ}$

(i) Calculate $\mathrm{f}\left(2\right)$

(ii) Write $\mathrm{f}\left(x\right)$ as a product of two algebraic factors.

Using the answer to (a)(ii), prove that there are exactly two real solutions to the equation

$-3y^6+8y^4-9y^2+10=0$

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

$\mathrm{f}\left(x\right)=x^4+a{x}^3-13x^2-38x-4b$

where $a$ and $b$ are constants.

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

Given that

$\frac{4x^4-37x^2+9}{2x-1}\equiv a{x}^3+b{x}^2+cx+d$

find the values of $a$, $b$, $c$ and $d$.

A function is defined as

$\mathrm{f}\left(x\right)=x^3+9x^2+rx+s$

where $r$ and $s$ are constants.

Given that

• $\mathrm{f}\left(2\right)=0$

• $\mathrm{f}(-1)=-54$

find the values of $r$ and $s$.

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

Given that

$(ax+by)(2x+y)(x-3y)=8x^3+c{x}^{2}y+dx{y}^2-9y^3$

find the values of $a$, $b$, $c$ and $d$.

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

A square has a side length of $(5x-2y+3)$ units.

Find an expression for the length of the diagonal of the square, in terms of $x$ and $y$.

Give your answer in the form

$\sqrt{a{x}^2+bxy+cx+d{y}^2+ey+f}$

where $a$, $b$, $c$, $d$, $e$ and $f$ are constants to be found.

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

Given that

• $\mathrm{f}\left(2\right)=0$

• $\mathrm{f}(-3)=-240$

find the values of $r$ and $s$.

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

Given that $3$ is a root of the equation

$2x^3-x^2-11x-12=0$

show that the equation has no other real roots.

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. 

Expand and simplify an expression for the volume of the cuboid in terms of $x$ and $y$.

Figure 4 shows a sketch of part of the curve $C_1$ with equation

$y=2x^3+10x>0$

and part of the curve $C_2$ with equation

$y=42x-15x^2-7x>0$

Verify that the curves intersect at $x=\frac{1}{2}$

The curves intersect again at the point $P$

Using algebra and showing all stages of working, find the exact $x$ coordinate of $P$

In this question you must show detailed reasoning.

Solutions relying on calculator technology are not acceptable.

The curve $C_1$ has equation $y=8-10x+6x^2-x^3$

The curve $C_2$ has equation $y=x^2-12x+14$

Verify that when $x=1$ the curves $C_1$ and $C_2$ intersect.

The curves also intersect when $x=k$.

Given that $k<0$

use algebra to find the exact value of $k$.

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

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.

A function is defined as

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

Given that $(2x+3)$ is a factor of $\mathrm{f}\left(x\right)$, use algebra to express $\mathrm{f}\left(x\right)$ as the product of four linear factors.

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

Find $\mathrm{f}(-1)$ and $\mathrm{f}(-2)$.

Solve

$2x^4-15x^3-10x^2+105x+98=0$