Polynomials (Edexcel IAL Maths): Exam Questions

Use the factor theorem to verify that $(x-2)$ is a factor of $x^{3}-x^{2}-14x+24$.

Divide $x^{3}+6x^{2}+11x+6$ by $( x + 2 )$.

Given $f(x)=x^{3}-5x^{2}-2x+24$ has a root at $x=3$, fully factorise $f(x)$.

Use polynomial division to show that $(x-2)$ is a factor of $2x^{3}+3x^{2}-18x+8$.

Given $x=-2$ is a root of the function $f(x)=x^{3}-3x^{2}-8x+4$, fully factorise $f(x)$.

Given that $(x-4)$ is a factor of $x^{3}-kx^{2}-4x+16$, find the value of $k$.

Given that $x=\frac{1}{2}$ is a root of the function $f(x)=2x^{3}+(p^{2}+1)x^{2}-11x+4$, find the possible values of $p$.

Given that $(x+1)$ is a factor of $f(x)=x^{3}-5x^{2}+3x+9$, fully factorise $f(x).$

Sketch the graph of $y=f(x)$, labelling the coordinates of all points where the graph intersects the coordinate axes.

Find the remainder when $x^{3}-7x-6$ is divided by $(x+4).$

The function $f(x)$ is given by $f(x)=x^{4}+ax^{3}-13x^{2}-38x-4b,$ where $a$ and $b$ are constants.

Given that both $(x+1)and(x+2)$ are factors of $f(x),$ find the values $a$ of $b$ .

Divide $x^{3}-6x^{2}-9x+14$ by $(x-7)$.

$f(x)=2x^{3}-x^{2}-16x+15$

Find the remainder when $f(x)$ is divided by $(x-2)$.

Given that $(x+3)$ is a factor of $f(x)$, factorise $f(x)$ completely.

$f(x)=2x^{3}-3x^{2}-72x-35$

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

Hence factorise $f(x)$ completely.

Write down all the real roots of the equation $f(x)=0$.

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

Use the factor theorem to show that $(x+2)$ is a factor of $f(x)$.

Factorise $f(x)$ completely.

Write down all the real roots of the equation $f(x)=0$.

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

find the values of $r$ and $s$.

Factorise $f(x)$ completely.

Consider the function $f(x)=4x^{3}+6x^{2}-7x+2.$

(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 $(x-2)(ax^{2}+bx+c)+d,$ where $a,b,c$ and $d$ are constants to be determined.

The function $f(x)=2x^{3}-5x^{2}+ax+b$ has $(2x+3)$ as a factor, and when $f(x)$ 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 $(x-5)$.

$f(x)=x^{3}-28x+48$

Find the remainder when $f(x)$ is divided by $(x-3)$.

Given that $(x+6)$ is a factor of $f(x)$, factorise $f(x)$ completely.

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

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

Hence factorise $f(x)$ completely.

Write down all the real roots of the equation $f(x)=0$.

$f(x)=4x^{3}-7x-3$

Use the factor theorem to show that $(2x+1)$ is a factor of $f(x)$.

Factorise $f(x)$ completely.

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Write down all the real roots of the equation $f(x)=0$.

$f(x)=x^{3}+rx^{2}+sx-30$. Given that $f(2)=0$ and $f(-3)=-240$ :

find the values of $r$ and $s$.

Factorise $f(x)$ completely.

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

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

Divide $4x^{4}-37x^{2}+9$ by $(2x-1)$.

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

Find the remainder when $f(x)$ is divided by $(2x+3)$.

Given that $(x+2)$ is a factor of $f(x)$, factorise $f(x)$ completely.

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

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

Given that $(x-3)$ is a factor of $f(x)$, factorise $f(x)$ completely.

Hence show that the equation $f(x)=0$ 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.

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

Show that $f(-1)=0$ and $f(-2)=0$.

Hence, solve $f(x)=0$.