1.5 No 43

#89
by fauyiyuji67 - opened

๐Ÿ˜ƒ: Usequantifiersandlogicalconnectivestoexpressthefact
that every linear polynomial (that is, polynomial of de
gree 1) with real coefficients and where the coefficient of
x is nonzero, has exactly one real root.
๐Ÿค–: Sure! Here's an outline for a proof using quantifiers and logical connectives to show that every linear polynomial with real coefficients and a nonzero coefficient of $x$ has exactly one real root:

I. Introduction

  • Briefly explain what a linear polynomial is and why it's important to consider only polynomials of degree 1.
  • State the main result: every linear polynomial with real coefficients and a nonzero coefficient of $x$ has exactly one real root.

II. Quantifier-based argument

  • Use quantifiers to prove that there exists at least one real number $r$ such that $f(r) = 0$, where $f$ is the given linear polynomial.
    + Proof idea: assume that no real number satisfies $f(r) = 0$, then construct a new real number $r$ that does satisfy this equation.
    + Details: suppose, if possible, that no real number $r$ satisfies $f(r) = 0$. Then, for any real numb...

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