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

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

Find the value of $k$ and hence find all the solutions to the equation $f\left(x\right)=0$.

Sketch the graph of $y=f\left(x\right)$, clearly indicating the coordinates of any points where the graph meets the coordinate axes.

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 A$\left(-3,0\right)$, B$\left(\frac{1}{2},0\right)$ and C$\left(1,-12\right)$.

Given that the degree of $f$ is as small as possible, find an expression 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 $k$.

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 $f\left(x\right)=8x^3-24x^2-72x+385$.

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 three distinct 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 $r=0,1,\dots ,n$.

The graph of $y=f\left(x\right)$, shown below, passes through $A\left(0,18\right)$. 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 expression for $f\left(x\right)$.

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 one possible set of integer values of $m,n$ and $p$.

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 $q$.

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

Sketch the graph of $y=f\left(x\right)$. Label the coordinates of any points where the graph meets 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 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 the values of $p$ and $q$ such that $x^2+px+q$ is a factor of $f\left(x\right)$ for both values of $k$.

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 $a$.