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Polynomials Class 10 Notes – CBSE Board | Lakshya Institution Gurugram

Polynomials Class 10 Notes

This chapter is very important for CBSE Class 10 Board Exams. Polynomials form the foundation for Algebra and are directly connected with Quadratic Equations and Higher Mathematics.

Polynomial Definition

A polynomial in variable x is an expression of the form:

a₀ + a₁x + a₂x² + a₃x³ + ... + aₙxⁿ

Degree of Polynomial

The highest power of the variable in the polynomial is called its degree.

Example:
3x² + 5x − 2 → Degree = 2
7x³ − 4x + 9 → Degree = 3

Zeroes of a Polynomial

If p(x) = 0 for some value of x, then that value is called a zero of the polynomial.

Relationship Between Zeroes and Coefficients

For quadratic polynomial ax² + bx + c:

Sum of zeroes = −b/a
Product of zeroes = c/a

Solved Examples (Board Pattern)

Example 1

Find zeroes of x² − 5x + 6.

Solution:

Factorizing:

x² − 5x + 6 = x² − 2x − 3x + 6

= x(x − 2) − 3(x − 2)

= (x − 2)(x − 3)

Zeroes are 2 and 3.

Example 2 (Using Formula)

Find zeroes of 2x² − 7x + 3.

a = 2, b = -7, c = 3

Using quadratic formula:

D = b² − 4ac

D = 49 − 24 = 25

x = [7 ± 5] / 4

x = 12/4 = 3

x = 2/4 = 1/2

Zeroes are 3 and 1/2.

Division Algorithm for Polynomials

Dividend = Divisor × Quotient + Remainder

Example

Divide 2x³ + 3x² − x + 5 by x − 1.

Using synthetic division:

Remainder = 9

Previous Year Questions (PYQ)

PYQ 1

Find a quadratic polynomial whose sum and product of zeroes are 4 and 3 respectively.

Solution:

Required polynomial:

x² − (sum)x + product

x² − 4x + 3

PYQ 2

If α and β are zeroes of polynomial x² − 6x + 8, verify relationship between zeroes and coefficients.

Solution:

Factorizing:

(x − 2)(x − 4)

Zeroes = 2, 4

Sum = 6

Product = 8

Verified: −b/a = 6 and c/a = 8

Important 4–5 Mark Questions

  1. Find zeroes of 3x² − 5x − 2.
  2. Verify relationship between zeroes and coefficients for 4x² − 4x − 3.
  3. Divide 3x³ + x² − 2x + 5 by x − 2.
  4. Find quadratic polynomial whose zeroes are −3 and 5.
  5. Find remainder when x³ − 3x² + 4x − 2 is divided by x − 1.

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