Chapter 2: Polynomials

Overview

This page provides an Overview of Chapter Wise Notes for CBSE Class 10. Key topics covered: Exam Weightage & Blueprint, Polynomial Basics ★★★★★, Important Formulas 🔥🔥🔥, Solved Examples (Board Marking Scheme), Exam Strategy & Mistake Bank, Self-Assessment Mock Test (10 Marks). All content is aligned with the NCERT syllabus for Board Exam preparation.

Board Exam Focused Notes, PYQs, and Verification Methods

Exam Weightage & Blueprint

Total: 4-6 Marks

Polynomials 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 $ax^2 + bx + c$:

$$ \text{Sum } (\alpha + \beta) = \frac{-b}{a} $$
$$ \text{Product } (\alpha \beta) = \frac{c}{a} $$

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 $\alpha^2 + \beta^2$.