📘 Class 10 Maths Chapter 2: Polynomials – Notes & Summary

 

🔍 What is a Polynomial?

A polynomial is an algebraic expression that includes variables, constants, and exponents combined using mathematical operations.

✅ Example:

$$2x^2 + 3x + 1$$

This is a quadratic polynomial (degree 2).

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🔹 Types of Polynomials

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🎯 Zeroes of a Polynomial

The zeroes (or roots) of a polynomial are the values of $x$ that make the polynomial equal to zero.

✅ Example:
If $p(x) = x^2 - 4$,
Then the zeroes are:

$$x = 2,\ x = -2$$

Because $p(2) = 0$ and $p(-2) = 0$

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🔁 Relationship Between Zeroes and Coefficients

For a quadratic polynomial $ax^2 + bx + c$, if α and β are the zeroes:

• Sum of zeroes = $\alpha + \beta = -\frac{b}{a}$

• Product of zeroes = $\alpha \cdot \beta = \frac{c}{a}$

✅ Example:
For $x^2 - 5x + 6$,
Sum = 5, Product = 6 → Roots: 2 and 3

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🧮 Division Algorithm for Polynomials

If $p(x)$ and $g(x)$ are two polynomials, then:

$$p(x) = g(x) \cdot q(x) + r(x)$$

Where:

• $p(x)$ = Dividend

• $g(x)$ = Divisor

• $q(x)$ = Quotient

• $r(x)$ = Remainder

• Degree of $r(x)$ < Degree of $g(x)$

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✅ Quick Revision

• A polynomial is an expression like $x^2 + 3x + 2$

• Degree = highest power of variable

• Use factorization and division algorithm for solving

• Know the relation between zeroes and coefficients

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