Chapter 2: Polynomials
Board Exam Focused Notes, PYQs, and Verification Methods
Exam Weightage & Blueprint
Total: 4-6 MarksPolynomials is part of the Algebra Unit (20 Marks Total). It is a high-scoring, low-effort chapter.
| Question Type | Marks | Frequency | Focus Topic |
|---|---|---|---|
| MCQ | 1 | High | Graphs (No. of Zeroes) |
| Short Answer | 2 or 3 | High | Relation b/w Zeroes & Coefficients |
| Case Study | 4 | Medium | Parabolic Path Applications |
Polynomial Basics ★★★★★
Degree: The highest power of $x$ in $p(x)$ is called the degree of the polynomial.
| Type | Degree | General Form | Max Zeroes |
|---|---|---|---|
| Linear | 1 | $ax + b$ | 1 |
| Quadratic | 2 | $ax^2 + bx + c$ | 2 |
| Cubic | 3 | $ax^3 + bx^2 + cx + d$ | 3 |
Important Formulas 🔥🔥🔥
1. Relationship (Quadratic)
For zeroes $\alpha$ and $\beta$ of $ax^2 + bx + c$:
$$ \text{Sum } (\alpha + \beta) = \frac{-b}{a} $$
$$ \text{Product } (\alpha \beta) = \frac{c}{a} $$
$$ \text{Product } (\alpha \beta) = \frac{c}{a} $$
2. Forming a Polynomial
$$ p(x) = k [ x^2 - (\alpha + \beta)x + (\alpha \beta) ] $$
(where k is a non-zero constant)
Solved Examples (Board Marking Scheme)
Q1. Find zeroes of $x^2 - 2x - 8$ and verify relationship. (3 Marks)
Step 1: Factorization 1 Mark
$x^2 - 4x + 2x - 8 = x(x-4) + 2(x-4)$
$\Rightarrow (x+2)(x-4)$. Zeroes: $-2, 4$.
Step 2: Sum Verification 1 Mark
Sum $= -2 + 4 = 2$. Formula: $-(-2)/1 = 2$.
Step 3: Product Verification 1 Mark
Product $= -2 \times 4 = -8$. Formula: $-8/1 = -8$.
Exam Strategy & Mistake Bank
Mistake Bank 🚨
Sign Error: Forgetting the negative in $-b/a$. If $b$ is already negative, result becomes positive!
X-axis only: In graph questions, count only X-axis intersections. Don't count Y-axis!
Scoring Tips 🏆
Show Calculation: For 3M questions, explicitly write "Sum of Zeroes = ..." and "$-b/a = ...$" separately.
Identity Use: For $t^2 - 15$, use $a^2-b^2$ identity to get zeroes $\pm\sqrt{15}$.
Self-Assessment Mock Test (10 Marks)
Q1 (1M): The number of zeroes for a quadratic polynomial is exactly 2. (True/False?)
Q2 (2M): Find a quadratic polynomial whose zeroes are $1/4$ and $-1$.
Q3 (3M): Find zeroes of $4u^2 + 8u$ and verify relationship.
Q4 (4M): If $\alpha$ and $\beta$ are zeroes of $x^2 + 4x + 3$, find the value of $\alpha^2 + \beta^2$.