Chapter 2: Polynomials

Overview

This page provides comprehensive Chapter 2: Polynomials - Board Exam Notes. Class 10 Maths Chapter Wise Notes Notes. Comprehensive revision notes, formulas, and key concepts for CBSE Board Exams.

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)

3. Relationship (Cubic)
For zeroes $\alpha$, $\beta$, and $\gamma$ of a cubic polynomial $ax^3 + bx^2 + cx + d$:

Sum: $\alpha + \beta + \gamma = \frac{-b}{a}$
Sum of Products: $\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$
Product: $\alpha\beta\gamma = \frac{-d}{a}$

4. Division Algorithm for Polynomials
If $p(x)$ and $g(x)$ are any two polynomials with $g(x) \neq 0$, then we can find polynomials $q(x)$ and $r(x)$ such that:
$p(x) = g(x) \times q(x) + r(x)$
where $r(x) = 0$ or degree of $r(x) < \text{degree of } g(x)$.

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