Chapter 2: Polynomials

Overview

This page provides comprehensive Chapter 2: Polynomials – Board Exam Notes aligned with the latest CBSE 2025–26 syllabus. Covers zeros of polynomials (graphical & algebraic), relationship between zeros and coefficients of quadratic polynomials, solved board questions, and interactive quiz.

Zeros of Polynomials • Graphical Method • Zeros-Coefficients Relationship • Solved Board Questions

Exam Weightage & Blueprint

Total: 4-6 Marks

Polynomials falls under Unit II: Algebra (20 marks total). As per the latest syllabus, focus is on finding zeros graphically and algebraically, and verifying the relationship between zeros and coefficients of quadratic polynomials.

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

Key Formulas & Relationships

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)

📌 Remember the Signs:
Sum of zeros = $-b/a$ (note the negative sign!)
Product of zeros = $c/a$ (constant term / leading coefficient)

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

📈 Zeros from Graph (Graphical Method)

The number of zeros of a polynomial $p(x)$ = number of times the graph of $y = p(x)$ intersects the x-axis.

Linear ($ax + b$)

Graph is a straight line. Crosses x-axis at exactly 1 point.

→ 1 zero

Quadratic ($ax^2 + bx + c$)

Graph is a parabola. Can cross x-axis at 0, 1, or 2 points.

→ 0, 1, or 2 zeros

Key Cases for Quadratic Graph:

2 zeros: Parabola cuts x-axis at 2 points (Discriminant $D > 0$)
1 zero (repeated): Parabola touches x-axis at 1 point ($D = 0$)
0 zeros: Parabola does not touch x-axis at all ($D < 0$)

where Discriminant $D = b^2 - 4ac$

⚠️ Common Mistake: Students count y-axis intersections as zeros. Only x-axis intersections are zeros! A zero is where $p(x) = 0$, i.e., $y = 0$.

📝 More Solved Board Questions

Q2. Find a quadratic polynomial whose zeros are $3 + \sqrt{2}$ and $3 - \sqrt{2}$. 2 Marks

Sol. Sum of zeros = $(3+\sqrt{2}) + (3-\sqrt{2}) = 6$

Product of zeros = $(3+\sqrt{2})(3-\sqrt{2}) = 9 - 2 = 7$

$p(x) = x^2 - (\text{sum})x + (\text{product}) = x^2 - 6x + 7$

Q3. If $\alpha$ and $\beta$ are zeros of $2x^2 - 5x + 3$, find $\alpha^2 + \beta^2$. 3 Marks

Sol. Here $a=2, b=-5, c=3$

$\alpha + \beta = \frac{-b}{a} = \frac{5}{2}$, $\alpha\beta = \frac{c}{a} = \frac{3}{2}$

$\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = \frac{25}{4} - 3 = \frac{13}{4}$

Q4. If one zero of $x^2 - 6x + k$ is double the other, find $k$. 3 Marks

Sol. Let zeros be $\alpha$ and $2\alpha$.

Sum: $\alpha + 2\alpha = 6 \Rightarrow 3\alpha = 6 \Rightarrow \alpha = 2$

Product: $\alpha \cdot 2\alpha = k \Rightarrow 2\alpha^2 = k \Rightarrow k = 2(4) = 8$

🎯 Board Pattern: Identity $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$ is the most frequently tested. Also know: $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha+\beta}{\alpha\beta}$ and $(\alpha-\beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta$.

📋 Board Revision Checklist

✅ Zeros of $p(x)$ = x-axis intersections of graph of $y = p(x)$
✅ Max zeros of polynomial of degree $n$ = $n$
✅ Quadratic: Sum of zeros $= -b/a$, Product $= c/a$
✅ Forming polynomial: $x^2 - (\text{sum})x + (\text{product})$
✅ $\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$
✅ $1/\alpha + 1/\beta = (\alpha+\beta)/(\alpha\beta)$
✅ Always verify: Sum and Product match $-b/a$ and $c/a$

💡 Exam Tip:
In 3-mark questions, always write verification separately — “Sum of zeros = ... = $-b/a$ = ... ✅ Verified”. This earns full marks even if factorisation has a minor slip.

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. The graph of a quadratic polynomial is a:

2. If zeros of $x^2 + 7x + 10$ are $\alpha, \beta$, then $\alpha + \beta$ = ?

3. A quadratic polynomial whose graph does not cross the x-axis has:

4. If product of zeros of $3x^2 + kx - 9$ is $-3$, then $k$ = ?

5. The number of zeros a cubic polynomial can have is: