Chapter 2: Polynomials

Class 10 Mathematics — Complete Concept & Formula Guide

NCERT Aligned Board Exam Focus Geometrical Analysis

1. Fundamental Definitions & Degrees

A polynomial in $x$ is an algebraic expression consisting of variables and coefficients, involving non-negative integer exponents only.

Degree of a Polynomial: The highest power of $x$ in a polynomial $p(x)$ is called its degree.

Classification by Degree

Type of Polynomial	Degree	Standard Form	Examples
Linear	1	$ax + b \quad (a \neq 0)$	$2x - 3$, $\sqrt{3}x + 5$
Quadratic	2	$ax^2 + bx + c \quad (a \neq 0)$	$2x^2 - 3x - 4$, $y^2 - 2$
Cubic	3	$ax^3 + bx^2 + cx + d \quad (a \neq 0)$	$x^3 - 3x^2 + x - 1$, $2 - x^3$

Non-Polynomial Expressions: Expressions containing variable exponents that are negative, fractional, or under square roots (e.g., $\frac{1}{x-1}$, $\sqrt{x} + 2$, $\frac{1}{x^2+2x+3}$) are NOT polynomials.

2. Value and Zeroes of a Polynomial

Value of a Polynomial at $x = k$

If $p(x)$ is a polynomial in $x$, and $k$ is any real number, the value obtained by replacing $x$ by $k$ in $p(x)$ is denoted by $p(k)$.

Zero of a Polynomial

A real number $k$ is said to be a zero of a polynomial $p(x)$ if:

$$\text{If } p(k) = 0, \text{ then } k \text{ is a zero of } p(x).$$

For a Linear Polynomial ($ax + b$):

Setting $ax + b = 0 \implies x = -\frac{b}{a}$. Therefore, the unique zero of a linear polynomial is:

$$\text{Zero} = -\frac{\text{Constant term}}{\text{Coefficient of } x} = -\frac{b}{a}$$

3. Geometrical Meaning of Zeroes

The zeroes of a polynomial $p(x)$ are precisely the $x$-coordinates of the points where the graph of $y = p(x)$ intersects or touches the $x$-axis.

Linear Graph

1 Point of Intersection
Exactly 1 Zero

Parabola ($a > 0$)

2 Intersecting Points
2 Distinct Zeroes

Parabola ($a < 0$)

1 Touching Point
2 Coincident Zeroes (1 unique)

Cubic Graph

Up to 3 Points
At most 3 Zeroes

Key Theorem / Rule: A polynomial $p(x)$ of degree $n$ intersects the $x$-axis at at most $n$ points. Therefore, a polynomial of degree $n$ has at most $n$ zeroes.

4. Relationship Between Zeroes & Coefficients

A. Quadratic Polynomial: $p(x) = ax^2 + bx + c \quad (a \neq 0)$

If $\alpha$ and $\beta$ are the zeroes of $p(x)$, then $(x - \alpha)$ and $(x - \beta)$ are factors of $p(x)$.

Quadratic Formulas:

$$\text{Sum of zeroes } (\alpha + \beta) = -\frac{b}{a} = -\frac{\text{Coefficient of } x}{\text{Coefficient of } x^2}$$ $$\text{Product of zeroes } (\alpha\beta) = \frac{c}{a} = \frac{\text{Constant term}}{\text{Coefficient of } x^2}$$

Forming a Quadratic Polynomial when Zeroes are Given:
If $\alpha$ and $\beta$ are given (or if Sum $S = \alpha+\beta$ and Product $P = \alpha\beta$ are given): $$p(x) = k \left( x^2 - (\alpha + \beta)x + \alpha\beta \right) = k \left( x^2 - S x + P \right)$$ (where $k$ is any non-zero real constant)

B. Cubic Polynomial: $p(x) = ax^3 + bx^2 + cx + d \quad (a \neq 0)$

If $\alpha, \beta, \gamma$ are the zeroes of the cubic polynomial, the relationships are:

Cubic Formulas:

$$\alpha + \beta + \gamma = -\frac{b}{a} = -\frac{\text{Coefficient of } x^2}{\text{Coefficient of } x^3}$$ $$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{\text{Coefficient of } x}{\text{Coefficient of } x^3}$$ $$\alpha\beta\gamma = -\frac{d}{a} = -\frac{\text{Constant term}}{\text{Coefficient of } x^3}$$

5. Worked Examples & Solutions

Example 1: Finding Zeroes & Verifying Relationship (Quadratic)

Question: Find the zeroes of $x^2 + 7x + 10$ and verify the relationship between zeroes and coefficients.

Solution:

1. Factorise by splitting the middle term:
   $x^2 + 7x + 10 = x^2 + 5x + 2x + 10 = x(x + 5) + 2(x + 5) = (x + 2)(x + 5)$
2. Find Zeroes:
   $p(x) = 0 \implies x + 2 = 0$ or $x + 5 = 0 \implies x = -2$ or $x = -5$.
   So, $\alpha = -2$ and $\beta = -5$.
3. Verify Relationships:
   • $\text{Sum of zeroes } (\alpha + \beta) = (-2) + (-5) = -7$
     From formula: $-\frac{b}{a} = -\frac{7}{1} = -7 \quad \checkmark \text{ (Verified)}$
   • $\text{Product of zeroes } (\alpha\beta) = (-2) \times (-5) = 10$
     From formula: $\frac{c}{a} = \frac{10}{1} = 10 \quad \checkmark \text{ (Verified)}$

Example 2: Polynomial with Difference of Squares

Question: Find the zeroes of $x^2 - 3$ and verify the relationship.

Solution:

1. $x^2 - 3 = (x - \sqrt{3})(x + \sqrt{3})$ using $a^2 - b^2 = (a-b)(a+b)$.
2. Zeroes are $\alpha = \sqrt{3}$ and $\beta = -\sqrt{3}$.
3. Verification:
   • $\alpha + \beta = \sqrt{3} + (-\sqrt{3}) = 0 = -\frac{0}{1} = -\frac{b}{a} \quad \checkmark$
   • $\alpha\beta = (\sqrt{3})(-\sqrt{3}) = -3 = \frac{-3}{1} = \frac{c}{a} \quad \checkmark$

Example 3: Forming Quadratic Polynomial

Question: Find a quadratic polynomial, the sum and product of whose zeroes are $\frac{1}{4}$ and $-1$ respectively.

Solution:

1. Given: Sum $S = \frac{1}{4}$, Product $P = -1$.
2. Required polynomial form: $p(x) = k(x^2 - Sx + P)$
3. Substitute $S$ and $P$:
   $p(x) = k \left( x^2 - \frac{1}{4}x - 1 \right) = \frac{k}{4} (4x^2 - x - 4)$
4. Taking $k = 4$, one such polynomial is: $4x^2 - x - 4$.

6. Quick Formula & Revision Sheet

Concept	Formula / Expression	Key Remark
Linear Zero	$x = -\frac{b}{a}$	1 real zero
Quadratic Zeroes Sum	$\alpha + \beta = -\frac{b}{a}$	Based on $ax^2 + bx + c$
Quadratic Zeroes Product	$\alpha \beta = \frac{c}{a}$	Based on $ax^2 + bx + c$
Quadratic Formation	$k[x^2 - (\alpha+\beta)x + \alpha\beta]$	$k$ is a non-zero constant
Cubic Sum	$\alpha + \beta + \gamma = -\frac{b}{a}$	Based on $ax^3 + bx^2 + cx + d$
Cubic Sum of Pairwise Products	$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$	Taken 2 at a time
Cubic Product	$\alpha\beta\gamma = -\frac{d}{a}$	Note negative sign

Class 10 Mathematics Revision Guide — Chapter 2: Polynomials