Factor Theorem

For a linear divisor or factor \(ax\pm b\), assume \(a\ne0\).

• Given a polynomial \( \mathrm{f}(x)\) is divided by a linear factor \(ax - b\), then \( \mathrm{f} \left( \frac{b}{a} \right) = 0 \)
  This follows from the division algorithm, where
  \( \mathrm{f}(x) = (ax - b) \mathrm{Q}(x)\)
  where \( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder
  *Note there is no remainder from the division since \( \mathrm{f}(x)\) is divided by its factor
  Then,
  \( \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) \)
  \( \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 = 0 \)