Factors of Polynomials (Copy)

FACTORS OF POLYNOMIALS – QUICK REVISION

1. Remainder Theorem

• If f(x) is divided by x − a, the remainder is f(a).
• If f(a) = 0, then x − a is a factor.

Example:
f(x) = 2x³ − 3x² − 11x + 6
Test x = 2: f(2) = 2(8) − 3(4) − 11(2) + 6 = 16 − 12 − 22 + 6 = −12 → remainder ≠ 0 → not a factor.
Test x = 3: f(3) = 54 − 27 − 33 + 6 = 0 → remainder = 0 → (x − 3) is a factor.

2. Factor Theorem

• If f(a) = 0, then x − a is a factor.
• Useful for finding linear factors before factoring further.

3. Finding Factors of Cubics

Step-by-Step:

1. Check possible rational roots using Rational Root Theorem:
   Possible roots = ± factors of constant term ÷ factors of leading coefficient.
2. Test each root with Remainder Theorem.
3. Once a factor (x − a) is found, use long division or synthetic division to divide cubic by (x − a).
4. Factor the remaining quadratic using:
   • Factorisation (if factorable)
   • Quadratic formula:
     x = (−b ± √(b² − 4ac)) / 2a

4. Example:

Find factors of f(x) = x³ − 6x² + 11x − 6.

Possible roots: ±1, ±2, ±3, ±6
Test x = 1: f(1) = 0 → (x − 1) is a factor.
Divide f(x) by (x − 1) → quotient = x² − 5x + 6.
Factor quadratic: (x − 2)(x − 3).

Final factorisation: f(x) = (x − 1)(x − 2)(x − 3).
Roots: 1, 2, 3.

5. Solving Cubic Equations

• Factor completely using above method.
• Set each factor = 0 to find roots.
• Roots may be:
  • All real (3 distinct)
  • Real + 2 equal
  • Real + 2 complex (from quadratic part)

6. Common Mistakes Table

7. Comparison Table – Remainder vs Factor Theorem