# Determine if there exist rational number a and irrational number A such that A

Determine if there exist rational number a and irrational number $A$ such that ${A}^{3}+a{A}^{2}+aA+a=0$
You can still ask an expert for help

## Want to know more about Irrational numbers?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

pressacvt
For any integer $a$ except $0$ or $1$, the polynomial ${x}^{3}+a{x}^{2}+ax+a$ has no rational roots. Any rational root $A$ would have to be an integer (by Gauss's lemma, or the Rational Root Theorem). Now ${A}^{3}+a{A}^{2}+aA+a=0$ means
$a=-\frac{{A}^{3}}{{A}^{2}+A+1}=-A+1-\frac{1}{{A}^{2}+A+1}$which, if $A$ is an integer, is not an integer unless $A=0$ (corresponding to $a=0$) or $A=-1$ (corresponding to $a=-1$): otherwise ${A}^{2}+A+1=\left(A+1/2{\right)}^{2}+3/4>1$.