Overview
This chapter studies polynomial equations through the relation between their roots and coefficients (Vieta's formulas), the nature of roots, and methods to transform equations.
1. Relation Between Roots and Coefficients
Quadratic ax² + bx + c = 0, roots α, β:
α + β = −b/a , αβ = c/a
Cubic ax³ + bx² + cx + d = 0, roots α, β, γ:
α+β+γ = −b/a , αβ+βγ+γα = c/a , αβγ = −d/a
2. Nature of Roots
- Complex roots of a real-coefficient polynomial occur in conjugate pairs (a ± ib).
- Surd (irrational) roots occur in pairs (a ± √b).
- A polynomial of degree n has exactly n roots (counting multiplicity).
3. Descartes' Rule of Signs
Number of positive real roots = number of sign changes in f(x) (or less by an even number). For negative roots, examine f(−x).
4. Transformation of Equations
To form an equation whose roots are k times / reciprocal of / increased by the original roots, substitute accordingly (e.g. replace x by x/k for roots multiplied by k).
Worked Examples
Example 1 — Sum and product of roots
For 2x² − 5x + 3 = 0, find α + β and αβ.
α + β = −(−5)/2 = 5/2 ; αβ = 3/2
Example 2 — Using a known root
One root of x³ − 6x² + 11x − 6 = 0 is 1. Find the others.
Sum of roots = 6, product = 6. With α = 1:
β + γ = 5, βγ = 6 → t² − 5t + 6 = 0 → t = 2, 3
Roots: 1, 2, 3
Example 3 — Form a new equation
Roots of x² − 3x + 2 = 0 are α, β. Form the equation whose roots are 2α, 2β.
New sum = 2(α+β) = 6 ; new product = 4αβ = 8
Equation: x² − 6x + 8 = 0
Example 4 — Descartes' rule
Discuss real roots of f(x) = x³ − 3x² + 4x − 5.
Signs of f(x): + − + − → 3 sign changes → 3 or 1 positive roots
f(−x) = −x³ − 3x² − 4x − 5 : all negative → 0 sign changes → 0 negative roots
Common Mistakes
- Forgetting the sign in α+β = −b/a.
- Assuming all roots are real (complex roots come in pairs).
- Miscounting sign changes in Descartes' rule.
- Wrong substitution while transforming equations.
Exam Tips & Revision Checklist
- Memorise Vieta's formulas for quadratic and cubic.
- Use a known root to reduce the degree (factor it out).
- Practised: sum/product of roots, reducing a cubic, transforming roots, Descartes' rule.
Frequently Asked Questions
For ax³ + bx² + cx + d = 0 with roots α, β, γ: α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a.