Polynomials

• How to Solve an Equation with Exponential and Polynomial Terms
  Learn when mixed exponential-polynomial equations have exact solutions, how to locate all real roots, and how to approximate them reliably.
• What Are the Conditions for Eisenstein’s Criterion?
  Eisenstein’s criterion requires a prime to divide every nonleading coefficient but not the leading coefficient or the constant term squared.
• How to Find the Real and Imaginary Parts of a Polynomial
  Find a polynomial’s real and imaginary parts by substituting z = x + iy, expanding, and collecting terms; verify the result with SymPy.
• Find All Irreducible Monic Cubics over the Field with Three Elements
  Over the field with three elements, a monic cubic is irreducible exactly when it has no roots; testing 0, 1, and 2 gives eight polynomials.
• Find All Irreducible Polynomials of Degrees 2 and 3 in F₂[x]
  The irreducible polynomials in F₂[x] of degrees 2 and 3 are x²+x+1, x³+x+1, and x³+x²+1; a root test proves the list.
• What Does the Height of an Algebraic Equation Mean?
  The height of an algebraic equation usually measures coefficient size, but algebraic-number height is a distinct invariant; here are both definitions.