📐 Polynomials: Notes, Zeroes & Coefficient Relationships

📑 In This Chapter Guide (Table of Contents)

1. 1. Degree & Geometrical Meaning of Zeroes of a Polynomial
2. 2. Relationship Between Zeroes and Coefficients
3. 3. High-Scoring Symmetric Expressions of α and β
4. 💡 Frequently Asked Questions (FAQ)

1. Degree & Geometrical Meaning of Zeroes of a Polynomial

The highest power of variable x in a polynomial P(x) is called its degree:

• Linear Polynomial (Degree 1): P(x) = ax + b (a ≠ 0). Graph is a straight line; intersects X-axis at exactly 1 point (Zero: x = -b/a).
• Quadratic Polynomial (Degree 2): P(x) = ax² + bx + c (a ≠ 0). Graph is a U-shaped curve called a Parabola. Opens upwards if a > 0, opens downwards if a < 0. Intersects X-axis at at most 2 points.
• Cubic Polynomial (Degree 3): P(x) = ax³ + bx² + cx + d (a ≠ 0). Intersects X-axis at at most 3 points.

Geometrical Zero Rule: The total number of zeroes of y = P(x) is equal to the exact number of points where the curve intersects the X-axis.

2. Relationship Between Zeroes and Coefficients

Let α (alpha) and β (beta) be the two zeroes of the quadratic polynomial P(x) = ax² + bx + c:

Forming a Quadratic Polynomial when Zeroes are Given:

If sum S = α + β and product P = α · β are known, the polynomial is given by:

3. High-Scoring Symmetric Expressions of α and β

Board exams frequently ask students to evaluate algebraic combinations without calculating α and β individually:

• α² + β² = (α + β)² - 2αβ = (-b/a)² - 2(c/a)
• (α - β)² = (α + β)² - 4αβ ⟹ α - β = √[(α + β)² - 4αβ]
• 1/α + 1/β = (α + β) / (αβ) = (-b/a) / (c/a) = -b/c
• α/β + β/α = (α² + β²) / (αβ) = [(α + β)² - 2αβ] / (αβ)
• α³ + β³ = (α + β)³ - 3αβ(α + β)

💡 Frequently Asked Questions (FAQ)

📚 Related Study Guides & Notes

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