3
votes
8answers
202 views

How do I solve $x^5 +x^3+x = y$ for $x$?

I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$? The explicit question out of the ...
2
votes
1answer
135 views

Finding the multiplicative inverse of an element in $\mathbb Q[x]/(x^3-2)$

I have a problem here that asks: "Express the multiplicative inverse of $1+2^{1/3}-3\cdot2^{2/3}$ as $a_0+a_1\cdot2^{1/3}+a_2\cdot2^{2/3}$." I believe they are asking us to find it by utilizing the ...
0
votes
0answers
53 views

Conversion of roots of a polynomial

I'm wondering, given a polynomial $P(x)$ with roots $r_i (1\le i\le n)$, how to determine the polynomial $Q(x)$ such that its roots are $r'_i=f(r_i)$. For example, if $P(x)=x^2-x-6=(x-3)(x+2)$ and ...
0
votes
1answer
34 views

Inverse function without the original function

I am going through this paper, 'Certifiable Quantum Dice Or, True Random Number Generation Secure Against Quantum Adversaries' by Vazirani and Vidick. In 'Our results' section on the page 2, it says: ...
0
votes
1answer
61 views

Find x in polynomial given value of inverse

I'm studying for a test and this question has me really stumped: $f(x) = 2x^3+5x+3$. Find x if $f^{-1}(x) = 1$ I don't know how I am supposed to figure out the inverse of this polynomial. I used ...
6
votes
1answer
105 views

My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible

Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial. Here's my ...
0
votes
1answer
649 views

Method of finding inverse of a Matrix using minimal polynomials

Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is ...
1
vote
2answers
69 views

Find the inverse of $\alpha^{38}$ in $\mathbb F = \mathbb Z_2[x]/\left<x^4+x+1\right>$

Let $\alpha$ be a root of $x^4+x+1$ and we are given some powers of $\alpha$ as linear combinations of $1,\alpha,\alpha^2$ and $\alpha^3$ $\alpha^4=\alpha+1$ $\alpha^5=\alpha^2+\alpha$ ... (the rest ...
4
votes
3answers
5k views

Inverse function of a polynomial

What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...
0
votes
2answers
534 views

Inverse function

Do you know how I could compute the inverse function of the following polynomial on $[0,1]$? $f(x)=\alpha x^3-2\alpha x^2+(\alpha+1) x$ (where $\alpha$ is within $]0,3[$)
2
votes
1answer
2k views

Homework Help - AP Calculus - Inverse of Polynomial

I know it is a simple problem but I am having trouble. Here is what I have so far: Let $f(x) = x^5 + 2x^3 + x - 1$ a) Find $f(1)$ and $f'(1)$ I have a) done. $f(1)$ is $3$ and $f'(1)$ is ...