Factor & Remainder Theorem Applications

This lesson covers applications of the factor and remainder theorems as required by the AQA A-Level Mathematics specification (7357). Building on the fundamentals covered in Lesson 1, this lesson focuses on using these theorems to solve cubic equations, find unknown coefficients, and sketch polynomial curves. These are skills that are frequently examined and essential for success at A-Level.

---

Solving Cubic Equations

A cubic equation has the form ax³ + bx² + cx + d = 0 (where a ≠ 0). Cubics always have at least one real root (since the graph must cross the x-axis at least once). The strategy for solving is:

1. Use the factor theorem to find one root by testing integer values.
2. Divide by the corresponding linear factor to obtain a quadratic.
3. Solve the quadratic (by factorising, completing the square, or using the formula).

Which Values to Test?

If f(x) = ax³ + bx² + cx + d, test values that are factors of d/a (the constant term divided by the leading coefficient). This is known as the rational root theorem.

For monic cubics (a = 1), test factors of d: ±1, ±2, ±3, etc.