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Grade 10 Math Lab: Proof & Generalization FREE
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Polynomial Equations & Roots
Learning Objectives
- Find real and complex roots
- Apply the Factor Theorem
- Use synthetic division
Key Concepts
Factor Theorem
If f(c) = 0, then (x - c) is a factor of f(x)
Synthetic Division
Shortcut for dividing a polynomial by (x - c)
Theory
**Factor Theorem:** f(c) = 0 ⟺ (x - c) is a factor.
**Rational Root Theorem:** Possible rational roots are ±(factors of constant) / (factors of leading coefficient).
**Synthetic Division Steps:**
1. Write coefficients in a row
2. Bring down first coefficient
3. Multiply by c, add to next coefficient
4. Repeat until done
Examples
Factor x³ - 6x² + 11x - 6
Solution: Test x=1: f(1)=0. Divide: (x-1)(x²-5x+6) = (x-1)(x-2)(x-3)