Polynomials (Cambridge International AS Mathematics): Exam Questions

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

Factorise

(i) $4 x^{2} - 4 x - 15$

(ii) $3 x^{3} + 11 x^{2} - 4 x$

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

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

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

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

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

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

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

Sketch the graph of y = \text{f} \left(x\right), labelling the coordinates of all points where the graph intersects the coordinate axes.

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

The polynomial \text{f} \left(x\right) is defined by \text{f} \left(x\right) = x^{4} + a x^{3} - 13 x^{2} - 38 x - 4 b, where $a$ and $b$ are constants.

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

The function \text{f} \left(x\right) is given by \text{f} \left(x\right) = 2 x^{3} + 7 x^{2} - 4 x. Show that \text{f} \left(x\right) = x \left(2 x - 1\right) \left(x + 4\right).

Hence, or otherwise, write down the real solutions to the equation \frac{\text{f} \left(x\right)}{x + 1} = 0.

The function \text{f} \left(x\right) is given by \text{f} \left(x\right) = x^{3} - 4 x^{2} - 7 x + 10. Work out \text{f} \left(1\right) and hence write down a factor of \text{f} \left(x\right).

Work out \text{f} \left(x\right) \div \left(x + 2\right).

Write \text{f} \left(x\right) in the form \left(x + a\right) \left(x + b\right) \left(x + c\right) where $a , b$ and $c$ are integers to be found.

Find the remainder when $x^{3} + 2 x^{2} - 5 x + 8$ is divided by \left(x - 3\right).

Given that \left(x^{2} - 8 x - 20\right) \div \left(x - 2\right) = A x + B + \frac{C}{x - 2}, where $A , B$ and $C$ are integer constants, in terms of $A , B$ and/or $C$ as appropriate:

(i) write down the divisor,

(ii) write down the quotient,

(iii) write down the remainder.

Find the values of $A , B$ and $C$.

The function $\text{f}$ is defined by \text{f} \left(x\right) = x^{2} + a x + b for $x \in ℝ$, where $a$ and $b$ are constants.

Given that \text{f} \left(3\right) = 0 and \text{f} \left(- 8\right) = 0, find the values of $a$ and $b$.

Use the factor theorem to show that \left(x - 2\right) is a factor of the function \text{f} \left(x\right) = x^{3} - 2 x^{2} - 4 x + 8.

Hence, or otherwise, express \text{f} \left(x\right) as a product of three linear factors.

Sketch the graph of y = \text{f} \left(x\right), labelling any points where the graph intersects the coordinate axes.

Divide $x^{3} - 19 x - 30$ by \left(x - 5\right).

For a polynomial \text{f} \left(x\right), the Remainder Theorem states that when \text{f} \left(x\right) is divided by \left(a x - b\right) the remainder is \text{f} \left(\frac{b}{a}\right).

Use the Remainder Theorem to find the remainder when $8 x^{3} + 6 x^{2} - x - 2$ is divided by \left(2 x + 1\right).

Work out the remainder when $6 x^{2} - x - 2$ is divided by \left(2 x + 1\right).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The polynomial \text{f} \left(x\right) is defined by \text{f} \left(x\right) = x^{3} + 9 x^{2} + r x + s.

Given that \text{f} \left(2\right) = 0 and \text{f} \left(- 1\right) = - 54, find the values of $r$ and $s$.

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

The polynomial \text{f} \left(x\right) is defined by \text{f} \left(x\right) = 3 x^{3} - 5 x^{2} - 4 x + 4.

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

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

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

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

Show that \left(2 x - 3\right) is a factor of $2 x^{3} - 13 x^{2} + 23 x - 12$.

Fully factorise $2 x^{3} - 13 x^{2} + 23 x - 12$.

Find all the real solutions to $2 x^{3} - 13 x^{2} + 23 x - 12 = 0$.

Given that \left(2 x - 1\right) is a factor of $2 x^{3} + x^{2} - 25 x + a$, find the value of $a$.

Given \left(x^{2} + 8 x - 4\right) \div \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 $x^{2} + 8 x - 4$.

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

Find \text{f} \left(x\right).

The result of dividing $x^{2} + a x - 5$ by \left(x + 1\right) is $x + 3 + \frac{d}{x+1}$. Find the values of $a$ and $d$.

The polynomial \text{f} \left(x\right) is defined by \text{f} \left(x\right) = 4 x^{3} + 6 x^{2} - 7 x + 2.

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

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

The polynomial \text{f} \left(x\right) is defined by \text{f} \left(x\right) = 2 x^{3} - 5 x^{2} + a x + b, where $a$ and $b$ are constants. It is given that \left(2 x + 3\right) is a factor of \text{f} \left(x\right), and that when \text{f} \left(x\right) is divided by \left(x - 2\right) the remainder is $7$.

Show that $a$ and $b$ satisfy the equations $2 a + b = 11$ and $3 a - 2 b = - 36$.

Hence find the values of $a$ and $b$.

Show that \left(x + 2\right) is a factor of $2 x^{3} - 3 x^{2} - 11 x + 6$.

Fully factorise $2 x^{3} - 3 x^{2} - 11 x + 6$.

Sketch the graph of $y = 2 x^{3} - 3 x^{2} - 11 x + 6$. Label any points where the graph crosses the coordinate axes.

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

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

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

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

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

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

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

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

Given that \left(4 x - 5\right) is a factor of $4 x^{3} - 9 x^{2} + a x + 30$, find the value of $a$.

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

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

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

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

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

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

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

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

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

\text{f} \left(x\right) = x^{3} + r x^{2} + s x - 30

Given that \text{f} \left(2\right) = 0 and \text{f} \left(- 3\right) = - 240, find the values of $r$ and $s$.

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

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

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

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

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

Given that \left(2 x - 5\right) is a factor of the function

\text{f} \left(x\right) = 2 x^{3} + k x^{2} - 11 x - 60

find the value of $k$ and fully factorise \text{f} \left(x\right).

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

Given that \left(x + 1\right) is a factor of $x^{3} - 4 x^{2} + x + 6$, sketch the graph of $y = x^{3} - 4 x^{2} + x + 6$. Label any points where the graph intersects the coordinate axes.

(There is no need to label any stationary points.)

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

Sketch the graph of y = \text{f} \left(x\right). Label any points where the graph crosses the coordinate axes.

It is given that

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

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

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

Explain, with an example, why the forms of \text{f} \left(x\right) and \text{g} \left(x\right) determined in parts (a) and (b) are not the only possible forms for those functions.

When $x^{3} + a x^{2} + 4 x - 1$ is divided by \left(x + 2\right) the quotient is $x^{2} - 4 x + 12$ and the remainder is $b$.

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

The polynomial p x^{3} + \left(5 p + 1\right) x^{2} + 5 q x - 2 q - 2, where $p$ and $q$ are constants, is denoted by \text{f} \left(x\right).

It is given that \left(x + 4\right) is a factor of \text{f} \left(x\right), and that when \text{f} \left(x\right) is divided by \left(x + 1\right) the remainder is $- 12$.

Find the values of $p$ and $q$.

Find the coordinates of the points of intersection between the curve with equation $y = x^{3} - x^{2} - 4 x + 4$ and the line with equation $y = 2 x + 4$.

On the same diagram, sketch the graphs of $y = x^{3} - x^{2} - 4 x + 4$ and $y = 2 x + 4$.

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

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

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

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

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

Show that \text{f} \left(- 1\right) = 0 and \text{f} \left(- 2\right) = 0.

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

When 2 x^{3} + \left(a + b\right) x^{2} + \left(a - b\right) x - 3 is divided by $x + 4$ the quotient is 2 x^{2} + \left(2 a + 3\right) x + \left(2 b - 5\right) and the remainder is $c$.

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

On the same diagram, sketch the graphs of $4 y = x^{3} - 5 x^{2} - 12 x + 36$ and $x + y - 6 = 0$.

Label the coordinates of any points of intersection between the two graphs.

Also label any points where the graphs intersect the coordinate axes.

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

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

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

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

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

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

Show that \left(9 x^{2} - 4\right) is a factor of $9 x^{4} - 40 x^{2} + 16$ and hence find all the real solutions to the equation $9 x^{4} - 40 x^{2} + 16 = 0$.

Show that \left(a x - 2\right) is a factor of 3 a x^{2} + \left(a - 6\right) x - 2.

Given that $x = - \frac{1}{a-4}$ is a root of 3 a x^{2} + \left(a - 6\right) x - 2, find the possible values of $a$.

\left(x - 2\right) and \left(x + 3\right) are factors of \text{f} \left(x\right), where \text{f} \left(x\right) = x^{4} - 9 x^{3} + 9 x^{2} + 85 x - 150.

Sketch the graph of y = \text{f} \left(x\right), labelling any points where the graph intersects the coordinate axes. (There is no need to label any stationary points.)

Sketch the graph of $y = 3 x^{3} + 2 x^{2} - 3 x + 10$, labelling any points where the graph intersects the coordinate axes.

Fully factorise $4 x^{3} + 17 x^{2} + 20 x + 4$.

Sketch the graph of $y = 4 x^{3} + 17 x^{2} + 20 x + 4$. Label any points where the graph crosses the coordinate axes.

On the same diagram, sketch the graphs of $y = x^{3} - 3 x^{2} - 6 x + 8$ and $y = \frac{3}{x^{2}}$.

Label any points where the graphs intersect the coordinate axes.

Write down the number of solutions to the equation

$x^{5} - 3 x^{4} - 6 x^{3} + 8 x^{2} = 3$.