Polynomial Functions (DP IB Analysis & Approaches (AA): HL): Exam Questions

Consider the function $f\left(x\right)=3x^3+p{x}^2+22x+q$,  where $p$ and $q$ are constants.  It is given that $(x^2-x+6)$ is a factor of $f\left(x\right)$.

Find the values of $p$ and $q$.

Find the roots of $f\left(x\right)$.

$\frac{3}{4}$ is a zero of the function $f\left(x\right)=4x^3-19x^2+kx-12,$ where $k$ is a constant.

As well as finding the value of $k$ , find all the solutions to the equation $f\left(x\right)=0$.

Sketch the graph of  $y=f\left(x\right)$.

The point $\left(\frac{7}{6},\frac{125}{108}\right)$  is a turning point on the graph $y=f\left(x\right)$.

Given that $f\left(x\right)=p$  has three distinct real solutions, where $p$ is a real constant, find the set of possible values of $p$.

The graph of $y=f\left(x\right)$ is shown below, where $f\left(x\right)$ is a polynomial function.
The graph passes through the points $\mathrm{A}\left(-3,0\right),\mathrm{B}\left(\frac{1}{2},0\right)$and $\mathrm{C}\left(1,-12\right)$.

Given that the degree of $f$ is as small as possible, find an equation for $f\left(x\right)$.

The graph is translated by the vector $\left(\begin{array}{c}k\\ 0\end{array}\right)$  to form the graph $y=g\left(x\right)$, where $k$ is a constant and $g\left(x\right)$ is a polynomial.

Given that $x$ is a factor of $g\left(x\right)$, find the possible values of $k$.

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.

For the function $f\left(x\right)=\left(3x-1\right)\left(x^2+x-1\right)\left(a{x}^2+bx+c\right)$, the sum of the roots is $\frac{1}{3}$ and the product of the roots is $-\frac{31}{36}$. 
Find all five roots of $f\left(x\right)$.

$\alpha$ and $\beta$ are non-real solutions of the equation $2x^2-\left(2k-3\right)x+2k=0$. 
Given that  ${\alpha }^2+{\beta }^2=\frac{9}{4}$ and $k\ne 0$,  find the value of $k$.

The function $f\left(x\right)=x^2-mx+3m-4$ has two integer solutions, one of which is double the other one. 

Find the value of $m$.

Consider the function $f\left(x\right)=p{x}^6+q{x}^4+r{x}^2+1$, where $p,q$and $r$ are real constants.

Show that if $\alpha$ is a zero of $f\left(x\right)$ then $-\alpha$  is also a zero.

Given that $\sqrt{5}$  and $-7-6\mathrm{i}$ are roots of the equation $f\left(x\right)=0$, find the value of $p$.

Let $f$ be a polynomial defined by $f\left(x\right)=8x^3-24x^2-72x+385.$

Use algebra to show that:

(i) $\left(2x+7\right)$ is a factor of $f\left(x\right)$,

(ii) $f\left(x\right)=0$ has exactly one real root.

Consider the function $g$ defined by $g\left(x\right)=f\left(x\right)+k$, where $k$ is a real constant.

Given that the equation $g\left(x\right)=0$  has exactly three real roots, find the set of possible values of $k$.

Consider the function $f\left(x\right)=2x^6-5x^5+p{x}^4+q{x}^3-2x^2+20x-8$, where $p$ and $q$ are constants.  It is given that $(x^2-x-2)$ is a factor of $f\left(x\right)$.

Show that $p=8$  and find the value of $q$.

Given that $-2\mathrm{i}$ is a root of $f$, find all of the roots of the equation $f\left(x\right)=0.$  .

Consider the function

$f\left(x\right)=\sum _{r=0}^n{a}_r{x}^r$ 

where $a_r\in \mathrm{ℝ}$ for $r=0,1,...,n$. 

The graph of $y=f\left(x\right)$, shown below, passes through $\mathrm{A}\left(0,18\right)$.  The roots of $f\left(x\right)$ are $\frac{3}{2},-1,\mathrm{i}$ and $-\mathrm{i}$.

Explain why $n$ must be even.

Given that $n$ is as small as possible, find an equation for $f\left(x\right)$.

A polynomial function $f$ is defined by $f\left(x\right)=k(m-x{)}^3(n-x)²$ where $k,m$ and $n$ are positive constants with $n>m$.

Sketch the graph of $y=f\left(x\right)$. Label the coordinates where the graph crosses the coordinate axes.

Determine the maximum number of distinct real solutions to the equation $f\left(x\right)=p$, where $p$ is a real constant.

Consider the function $g\left(x\right)=f\left(ax+b\right)$, where $a$ and $b$ are positive constants. The points $\left(0,0\right)$ and $\left(1,0\right)$ lie on the graph $y=g\left(x\right)$.

Find $a$ and $b$ in terms of $m$ and $n$.

Consider the function $g$ defined by $g\left(x\right)=a{x}^3+4b{x}^2+(4a-3)x-3b$,  where $a,b\in \mathrm{ℝ}$ are constants.

Given that $\left(x-3\right)$ is a factor of $g\left(x\right)$, and that the sum of the roots of the equation $g\left(x\right)=0$ is 5, 

(i) find the values of $a$ and $b$, and

(ii) hence factorise $g\left(x\right)$ fully.

Consider the function $f$  defined by  $f\left(x\right)=\left(2x^3+9x^2+4x-15\right)\left(m{x}^2+nx+p\right)$,  where $m,n$and $p$ are real constants. 

It is given that the sum of the roots of the equation  $f\left(x\right)=0$ is  $-\frac{41}{6}$,  and that the product of the roots is $\frac{25}{2}$. 

Find a set of values for $m,n$and $p$ that satisfies the above conditions, such that $m,n,p\in \mathrm{\mathbb{Z}}$. .

The equation $x^2+(k-1)x-2k=0,k\in \mathrm{ℝ}$ has non-real roots $\alpha$ and $\beta$ where  ${\alpha }^3+{\beta }^3=5$. 

Find the value of k.

The equation $x^2+px+q=0$  has roots ${\alpha }^3$ and ${\beta }^3$.

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

Consider the polynomial function defined by

$f\left(x\right)=\sum _{r=0}^5{a}_r{x}^r,$ 

Where the $a_r$ are real constants. The function has the property that $f\left(-x\right)=-f\left(x\right)$ for all values of $x$. 

Show that  $a_0=a_2=a_4=0.$

Given that $-2+3\mathrm{i}$ is a root of the equation $f\left(x\right)=0,$

(i) show that $2-3\mathrm{i}$ is also a root of $f\left(x\right)=0$, and 

(ii) hence find the values of $a_1$ and $a_3$in terms of $a_5$.

Consider the polynomial function $f\left(x\right)=x^4+a{x}^3+b{x}^2+cx+d$,  where $a,b,c,d\in \mathrm{ℝ}$.

Two distinct roots of $f\left(x\right)=0$ are given by $k+k^2\mathrm{i}$ and $k^2+k\mathrm{i}$,  where $k$ is a real constant.

The remainder when $f\left(x\right)$ is divided by $x$ is 8100.

(i) Find the two possible values of $k$.  

(ii) Hence find real values for $p$ and $q$ such that $(x^2+px+q)$ is guaranteed to be a factor of $f\left(x\right)$.

Given that $a=-12$ , find the values of $b$ and $c$.

The polynomial function $f$ is defined by

$f\left(x\right)=2a{x}^3+(4+2a-a^2)x^2-(6+2a+a^2)x+3a$ 

where $a\ne 0$ is a real constant. 

The graph of $y=f\left(x\right)$ only intersects the $x$-axis at the point $\left(\frac{a}{2},0\right)$.

By considering the sum of the roots, use proof by contradiction to show that $f\left(x\right)=0$ has two non-real roots.

Find the set of possible values of $a$.