Is a polynomial of degree 3 with irrational roots possible? It is easy to give an example of a polynomial of degree 3 with integer coefficients having:   (a) three distinct rational roots,  (b) one rational root and two irrational roots.  But for a while I am trying to construct one that all its roots are irrational but I can't. It seems that it is not possible at all?   Also, can a polynomial with integer coefficients of degree 3 have two rational roots and one irrational root?
Cubic polynomial with three (distinct) irrational roots I am looking for an equation     $$x^3+ax^2+bx+c=0, \qquad a, b, c \in \Bbb Z,$$   of degree $3$ that has $3$ different roots.  For an equation of degree $2$ it is easy---for example $x^2-2=0$---but I can't find an example for degree $3$.  if there is a method for finding this kind of Equation I will be grateful.
Unsized array declaration in a struct How to print variables without spaces between values GLR Test for 2 samples from exponential distributions Why is a 1" sensor actually 13.2 × 8.8mm? I'm wondering how sensor sizes are calculated. The Sony RX100 reportedly has a sensor size of 1" which is 13.2 x 8.8 mm according to the press release and various sites.  How does that work? That's a pretty small inch.  The Wikipedia page on  has this chart:   which doesn't really help. (The 1" format is labelled Nikon CX. Chart by Moxfyre under the CC-BY-SA license.)   If full frame/35 mm refers to the horizontal size, what does 1" refer to? How can I use adb to send a longpress key event? What happens to the semiconductor in an LED when it is given voltage that it cannot handle? Is there a limit to the number of magic rings a character can wear on one hand? Can someone please explain how I can install  in the following ways?   By using the package manager? By downloading and installing a pre-built package? How to interpret decomposition plot and check for seasonality Lufthansa airline offers the &quot;Economy Flex&quot; fare. The listed advantages of this fare are:  Rebooking possible plus fare difference Refundable except 70 EUR plus fare difference  What does the &quot;plus fare difference&quot; mean? Imagine I buy a ticket for 200 €. Later I decide to cancel. They keep 70 Euro, so I get 130€ back. What's with the fare difference? Fare difference to what? As a comparison, the &quot;Economy classic&quot; states:  Rebooking: 70 EUR plus fare difference Refund not possible The question asks us to classify all distributions in $\mathbb{R}^2$, say $u$, such that $$(xy)u(x,y) = (x^2-y^2)u(x,y)= 0.$$ I am genuinely stuck, any help is extremely appreciated. Rebel flight suits include an ejection harness, Biggs tells Porkins to eject. This seems to suggest that Rebel fighters have an ejection system.  How would they operate for different fighters? Does the whole cockpit separate? Do they function like current aircraft ejection systems - if so what protects the pilot from the vacuum of space? Why does Hermione say she hadn't used a memory charm before? After the wedding, Hermione has to use a memory charm on two Death Eaters, where she states that she has never done it before, but she knows the theory. However, she states that before she left on her adventure, she made her parents forget her, and move to Australia. Is this a contradiction, or is there an explanation? How do I determine if an array is initialized in VB6? Passing an undimensioned array to the VB6's Ubound function will cause an error, so I want to check if it has been dimensioned yet before attempting to check its upper bound.  How do I do this? Can two PS4s log in to the same PSN account? Both myself and my boyfriend have PS4s.  Can I just get a single PSN account and sign both consoles into the same account, or do I need to get two separate accounts? Verb agreement with "what" This came up when reviewing a written English exercise, and I realized that I don't entirely trust my intuition on the answer.  Which of the following is correct?  Or are both valid?  (Or is it strongly advisable to rewrite the sentence in some other way?)   "I see what appears to be a man and his son." "I see what appear to be a man and his son." drawing different parts of a line in different colours How to draw different parts of a line in different colours. In the following code can each part of the line be drawn in different colour      \documentclass{beamer} \beamertemplatenavigationsymbolsempty \usepackage{tikz} \usepackage{pgfplots} \begin{document} \begin{frame}[fragile,t] \frametitle{} { \begin{tikzpicture} \draw [thick, green] (2.3,1.4) +(-27:2.2cm)-- +(153:2.2cm) +(42:1.88cm) -- +(-138:1.88cm); \end{tikzpicture} } \end{frame} \end{document} Are there benefits of passing by pointer over passing by reference in C++? What are the benefits of passing by pointer over passing by reference in C++?  Lately, I have seen a number of examples that chose passing function arguments by pointers instead of passing by reference.  Are there benefits to doing this?  Example:  func(SPRITE *x);   with a call of  func(&amp;mySprite);   vs.  func(SPRITE &amp;x);   with a call of  func(mySprite); Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials I am reading Gathmann's free online notes on Algebraic Geometry. One exercise asks to show that     "Every affine variety in $\mathbb A^n$ consisting of finitely many points can be written as the zero locus of $n$ polynomials".    There is a hint says "interpolation". I don't know how to start with the hint.   If $n=2$, we can use interpolation to get 1 polynomial for finitely many points. But we need to show 2 polynomials instead. I am also not sure how to apply interpolation for higher dimensions. Anyone can help? Thank you! Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$ Show that    $$1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$$   So, I use Fermat's little theorem, that is if $p$ does not divide $a$, then $a^{p-1}$ is congruent to $1$ (mod $p$). But I'm still confused. If $1^{p-1}$ is congruent to $1$ (mod $p$), $2^{p-1}$ is congruent to $1$ (mod $p$), and $p$ does not divide $p-1$, so $(p-1)^{p-1}$ is congruent to $1$ (mod $p$). How does that equal $-1$ (mod $p$)?