📘 Polynomials Class 10 Notes (Chapter 2) – Easy & SEO Optimized

🔷 Introduction to Polynomials

Polynomials are important in algebra. In simple words, a polynomial is an expression made using variables, numbers, and mathematical operations like addition and multiplication.

For example:
$P(x) = 2x^2 + 5x + 1$P(x)=2×2+5x+1

🔹 What is a Polynomial?

A polynomial is an algebraic expression that can take any real value of x.

In other words, it is a combination of variables and constants.

🔹 Basics of Algebra

In algebra, two main parts are used:

• Variables → letters like x, y, z
• Constants → fixed numbers like 2, 5, 10

Therefore, algebra helps us represent unknown values easily.

🔹 Terms in Algebra

Each part of an expression is called a term.

For example:
3x, 9x², 5xy

Moreover, terms are connected using addition or subtraction.

🔹 Like and Unlike Terms

✔ Like Terms

Like terms have the same variables and powers.

For example:
3x² and 2x²

✔ Unlike Terms

Unlike terms have different variables or powers.

For example:
11x² and 2x³

Hence, like terms can be added, but unlike terms cannot.

🔹 Algebraic Expressions

An algebraic expression is formed by combining terms.

For example:

• 2x² + 5
• 7x + 8xy
• 9x² + y²
• 3x² + 4x + 5

As a result, expressions help in solving mathematical problems.

🔹 Examples of Polynomials

Some common polynomial examples are:

• $P(x) = 2x^2 + 5$P(x)=2×2+5
• $P(x) = 2x^2 + 7x + 6$P(x)=2×2+7x+6

🔹 Equations

An equation is formed when an expression is equal to zero.

For example:

• 3x + 5 = 0
• 9x² + 8x + 5 = 0

Therefore, equations are used to find unknown values.

🔹 Zeroes of a Polynomial

Zeroes are the values of x that make the polynomial equal to zero.

If α and β are zeroes of:$$P(x) = ax^2 + bx + c, \quad a \neq 0$$P(x)=ax2+bx+c,a=0

Then:

• Sum of zeroes:

$$\alpha + \beta = -\frac{b}{a}$$α+β=−ab​

• Product of zeroes:

$$\alpha \beta = \frac{c}{a}$$αβ=ac​

Additionally, $(x – α)$(x−α) and $(x – β)$(x−β) are factors of the polynomial.

🔷 Types of Polynomials

🔹 According to Number of Terms

✔ Monomial

A polynomial with one term.
Example: $2x^2$2×2

✔ Binomial

A polynomial with two terms.
Example: $2x^2 + 5$2×2+5

✔ Trinomial

A polynomial with three terms.
Example: $2x^2 + 4x + 5$2×2+4x+5

🔹 According to Degree

✔ Linear Polynomial (Degree 1)

Example: $2x + 3$2x+3

✔ Quadratic Polynomial (Degree 2)

Example: $x^2 – 2x + 1$x2−2x+1

✔ Cubic Polynomial (Degree 3)

Example:
$3x^3 + 5x^2 – 2x + 1$3×3+5×2−2x+1

Thus, polynomials are classified based on their highest power.