My assignment is to translate mathematical statements into formula of predicate logic. But before I can write formulas about these statements, I'm really confused about their meaning.
Given that the mathematical notation of a quadratic polynomial with leading coefficient 1 is $P(x) = a_0 + a_1x + x^2$
(1) " $w$ is a root of infinitely many quadratic polynomials with leading coefficient 1 " ==> if I want to write this into a logical statement, should my formula says something to the negation of the statement ? i.e. "there are infinitely many quadratic polynomials with leading coefficient 1 where $w$ is not a root" ?
Or should I say : " For all the coefficients $a_0, a_1$ such that $P(x) = x^2 + a_1x + a_0$ is $0$ when I plug in $x$, then I can always find the other two coefficients $b_0, b_1$ (different than $a_0, a_1$) such that when I replace them with the $a_0, a_1, P(x)$ is still $0$ at $x$" ?
What does the statement actually mean?
(2) " $w$ is a root of polynomials of arbitrarily large degree with leading coefficient 1." ==> Does the statement mean "there are infinitely many polynomials of degree $n$ with leading coefficient 1, such that $w$ is a root ?" or it just means that "if I can find a polynomial of degree $n$ such that $w$ is a root, then I can find a different polynomial of degree $n$ where $w$ is also a root" ?
Would someone please help me on this? I kept thinking about them for couple days, and I'm still lost >_< Thank you in advance :)