Inequality

Union (U)

• The union of two inequalities includes all the values that satisfy at least one of the inequalities.

• Think of it as: "this OR that (or both)".

  • Solve:
    x<−2orx>3x < -2 \quad \text{or} \quad x > 3x<−2orx>3

    Union: Combine both sets:
    → (−∞,−2)∪(3,∞)(-\infty, -2) \cup (3,∞)(−∞,−2)∪(3,∞)

    This means any value less than -2 or greater than 3 is part of the solution.

Intersection ( ∩ )

• The intersection includes only the values that satisfy both inequalities at the same time.

• Think of it as: "this AND that".

  • Solve:
    x>−2andx<3x > -2 \quad \text{and} \quad x < 3x>−2andx<3

    Intersection:
    → (−2,3)(-2, 3)(−2,3)

    Only values between -2 and 3 (not including -2 and 3 themselves) work for both inequalities.

Prime Polynomials

• A prime polynomial (also called an irreducible polynomial) is a polynomial that cannot be factored into simpler polynomials with coefficients in a given set (like real numbers or integers), other than by multiplying by 1 or -1.

  • Over the real numbers:
    x2+1x^2 + 1x2+1 is prime, because it cannot be factored using real numbers.

  • Over the complex numbers:
    x2+1=(x+i)(x−i)x^2 + 1 = (x + i)(x - i)x2+1=(x+i)(x−i), so it's not prime in that context.