A polynomial with integer coefficients is an expression of the form:
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
where $a_n$, $a_{n-1}, \ldots, a_1, a_0$ are integers and $a_n$ is not equal to $0$.
a zero of the polynomial is a $c \in \mathbb{R}$ such that $f(c)=0$
A real number is said to be algebraic if it is a zero polynomial with integer coefficients
1) Show that every rational number is algebraic
2) Show that if $a$, $b$ and $k$ are positive integers, then the $k$-th root of $a/b$ is algebraic
I don't even know where to start on this. What is a zero of a polynomial with integer coefficients?
Best Answer
The rational number $5/7$ is a zero of the polynomial $7x+(-5)$. We have $n=1$, $a_1=7$, $a_0=-5$.
So try showing that works with every rational number.