2.12 Polynomial functions questions for IB Maths AA

Show that $\tan 3\theta \equiv {\displaystyle \frac{3\tan \theta -{\tan }^3\theta }{1-3{\tan }^2\theta }}$, for all values of $\theta$ for which $\tan \theta$, $\tan 2\theta$ and $\tan 3\theta$ are defined.

Hence show that $t=\tan {\displaystyle \frac{\pi }{12}}$ satisfies the equation $t^3-3t^2-3t+1=0$.

Show that $t=-1$ is a root of $t^3-3t^2-3t+1=0$, and hence find the other two roots in exact form.

Hence find the exact value of $\tan {\displaystyle \frac{\pi }{12}}$, justifying your answer.

Find all the values of $\theta$, where $-{\displaystyle \frac{\pi }{2}}<\theta <{\displaystyle \frac{\pi }{2}}$, for which $\tan 3\theta =1$.

By considering the sum of the roots of $t^3-3t^2-3t+1=0$, show that $\tan {\displaystyle \frac{\pi }{12}}+\tan {\displaystyle \frac{5\pi }{12}}=4$.

Consider the case $c=\text{i}$. Find $z_1$, $z_2$, $z_3$ and $z_4$, giving your answers in the form $a+b\text{i}$, where $a,b\in \mathbb{Z}$.

Hence, for $c=\text{i}$, find $z_{2026}$. Justify your answer.

Consider the case $c=1-\text{i}$.

Find $z_2$ and $z_3$.

Find $\mid z_4\mid$.

For a general value of $c$, find $z_2$ and $z_3$ in terms of $c$.

Show that the non-zero values of $c$ for which $z_3=0$ satisfy $c^3+2c^2+c+1=0$.

Hence find all the non-zero values of $c$ for which $z_3=0$.

Use a double angle identity to show that $T_2(x)=2x^2-1$.

By expanding $\cos ((n+1)\theta )$ and $\cos ((n-1)\theta )$, show that $\cos ((n+1)\theta )+\cos ((n-1)\theta )\equiv 2\cos \theta \cos n\theta$.

Hence show that $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$, for $n\in {\mathbb{Z}}^{+}$.

Hence find $T_3(x)$ and $T_4(x)$.

Use your GDC to sketch the graph of $y=T_4(x)$ for $-1\le x\le 1$, and write down the range of $T_4$ on this interval.

Write down the leading coefficient of $T_n(x)$ for $n=1,2,3,4$, and hence suggest an expression for the leading coefficient of $T_n(x)$, for $n\in {\mathbb{Z}}^{+}$.

Prove by mathematical induction that, for $n\in {\mathbb{Z}}^{+}$, $T_n(x)$ is a polynomial of degree $n$ with leading coefficient $2^{n-1}$.

Show that the $n$ roots of $T_n(x)=0$ are $x=\cos \left({\displaystyle \frac{(2k-1)\pi }{2n}}\right)$, for $k=1,2,\dots ,n$.

Hence find the exact value of $\cos {\displaystyle \frac{\pi }{8}}$.

By considering the product of the roots of $T_4(x)=0$, find the exact value of $\cos {\displaystyle \frac{\pi }{8}}\cos {\displaystyle \frac{3\pi }{8}}$.

Show that $c={\displaystyle \frac{6}{a^3}}$.

Write down $\mathrm{E}(X)$, giving a reason for your answer.

Show that $\mathrm{V}\mathrm{a}\mathrm{r}(X)={\displaystyle \frac{a^2}{20}}$.

It is given that $\mathrm{P}(X<1)={\displaystyle \frac{5}{32}}$.

Explain why $a>1$.

Show that $5a^3-96a+64=0$.

Hence find the value of $a$, justifying your answer.

Find the standard deviation of $10-5X$.

Find a quadratic factor of $p(z)$ with real coefficients.

Hence find the other two roots of $p(z)=0$.

Consider the case $p(x)=2x^3-5x^2+4x-7$.

Find $p^{\mathrm{\prime }}(x)$, $p^{\mathrm{\prime }\mathrm{\prime }}(x)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(x)$.

Find $p(0)$, $p^{\mathrm{\prime }}(0)$, $p^{\mathrm{\prime }\mathrm{\prime }}(0)$ and $p^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)$. Hence verify that ${\displaystyle \frac{p^{(k)}(0)}{k!}}$ is equal to the coefficient of $x^k$ in $p(x)$, for $k=0,1,2,3$.

Now consider the general polynomial $p$. Show that $p^{(k)}(0)=k!a_k$, for $k=0,1,\dots ,n$.

Hence write $p(x)$ in terms of $x$ and the values $p^{(k)}(0)$, for $k=0,1,\dots ,n$.

Use part (c) with $p(x)=(1+x{)}^n$ to show that the coefficient of $x^k$ in the expansion of $(1+x{)}^n$ is ${\displaystyle \frac{n!}{k!(n-k)!}}$.

Fix a value of $x$, and let $q(h)=p(x+h)$, a polynomial in $h$ of degree $n$.

By applying part (c) to $q$, show that

Hence show that ${\displaystyle \underset{h\to 0}{\lim }{\displaystyle \frac{p(x+h)-p(x)}{h}}=p^{\mathrm{\prime }}(x)}$.

Let $p(x)=x^4-2x^3$. Use the result in part (e) with $x=1$ to express $p(1+h)$ as a polynomial in $h$. Hence find the exact value of $p(1.1)$, and check your answer using your GDC.

The polynomial $p$ is said to have a repeated root at $x=a$ if $p(x)=(x-a{)}^{2}r(x)$ for some polynomial $r$.

Show that $p$ has a repeated root at $x=a$ if and only if $p(a)=0$ and $p^{\mathrm{\prime }}(a)=0$.

Find the values of $k$ for which the equation $x^3-3x+k=0$ has a repeated root. For each value of $k$, write down all the roots of the equation.