Remainder Theorem

• Given a polynomial \( \mathrm{f}(x)\) is divided by a linear expression \(ax - b\), then its remainder is \( \mathrm{f} \left( \frac{b}{a} \right) \)
  This follows from the division, where
  \( \mathrm{f}(x) = (ax - b) \mathrm{Q}(x) + \mathrm{R}\)
  where \( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder

\( \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) + \mathrm{R} \)
\( \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 + \mathrm{R} = \mathrm{R} \)

Remainder when divided by higher degree polynomial

• When \( \mathrm{f}(x) \) is divided by a polynomial of degree \(n\), its remainder will have a degree less than \(n\)
• For example when \( \mathrm{f}(x)\) is divided by a quadratic expression \(ax^2 + bx + c\), its remainder will be a linear expression \(dx + e\).
• In summary: