Chapter 2: Polynomials
Decoding Degrees, Graphing Zeroes, and Coefficient Relationships.
📘 Degree 1: Linear
Highest power of x is 1.
Examples: 2x - 3, √3x + 5.
If p(k) = 0, then ak + b = 0 ⇒ k = -b/a.
📗 Degree 2: Quadratic
Highest power of x is 2. The word comes from 'quadrate' meaning 'square'.
Example: 2x² + 3x - 5.
📙 Degree 3: Cubic
Highest power of x is 3.
Example: 3x³ - 2x² + x - 1.
📈 Geometrical Meaning of Zeroes
The zeroes of p(x) are precisely the x-coordinates of the points where the graph of y = p(x) intersects the x-axis.
Straight line. Intersects exactly ONCE at (-b/a, 0).
Parabola (U-shape). Intersects at MOST twice.
🔗 Zeroes & Coefficients Relation
For quadratic polynomial ax² + bx + c, with zeroes α and β:
Sum: α + β = -b / a
Product: αβ = c / a
⭐ Board Exam Super-Hacks
Building Polynomial from Zeroes
If you are given the sum (S) and product (P) of zeroes, the quadratic polynomial can be quickly formed using:
k[x² - (Sum)x + (Product)] or k(x² - Sx + P).
Graph Intersection Rule
A polynomial p(x) of degree n can intersect the x-axis at at most n points. Therefore, it has at most n zeroes. Count the x-axis touches to find the number of zeroes!
⚠️ Common Exam Traps
Trap 1: The Missing Minus Sign
When calculating the Sum of zeroes (α + β = -b/a), students often forget the negative sign. Always write the formula first to avoid missing it.
Trap 2: Recognizing Polynomials
Expressions with negative powers (e.g., 1/x - 1) or fractional powers (e.g., √x + 2) are NOT polynomials. Ensure all powers of x are whole numbers.