﻿ Class 10 NCERT Math Solution
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Introduction

Introduction of Polynomials

Polynomials = Poly (means many) + nomials (means terms). Thus, a polynomial contains many terms.
A Polynomial is a type of algebraic expression with many terms having variables and coefficients.
e.g., x, 2x² + 3, 3x³ + 2x² + x + 2, 4x⁴ + 3x³ + 2x² - x + 1 etc.

Monomials

Monomials = Mono (means one). Thus, a monomial contains only one term.
Polynomials having only one term are called monomials. e.g., x, 2x, 2x² etc.

Binomials

Binomials = Bi (means two). Thus, a binomial contains two terms.
Polynomials having only two terms are called binomials e.g., 2x + 5, 2x² + 3, 2x² + 3x etc.

Trinomials

Trinomials = Tri (means three). Thus, a trinomials contains three terms.
Polynomials having only three terms are called trinomials e.g., 4x⁴ + 2x² - 3, 2x² + 3x + 1 etc.

Degree of the polynomial or Exponents

The highest power of the variable in a polynomial is called degree of the polynomial or exponents.
The degree of a non-zero constant polynomial is zero. i.e., 2, 5 where, degree is 0. Because there is no any variable present.
e.g., 2x² + 3x + 1. Here, 2 is the highest degree of polynomial.

Variable

A variable is a number which is use in alphanumeric character. e.g., x, y, a, b etc.

Constant or Non-variable

A non-variable or constant is a number which is use in numeric values. e.g., 2, 3, 10 etc.

Terms

In the polynomial x² + 3x, the expressions x² and 3x are called the terms of the polynomial.

Coefficient

Multiplicative numerical or Constant value attached with a polynomial is called coefficient. e.g., 2x² + 3, - x + 1 etc.
Here, 2 is the coefficient of x², -1 is the coefficent of -x.

Constant Polynomials

In polynomials only constant or non-variable values are used i.e., called constant polynomials. e.g., 3, -5, 4 etc.

Zero Polynomial

The constant polynomial 0 is called the zero polynomial.

Linear Polynomials

A polynomial of degree one is called a linear polynomial. e.g., 4x - 3, 2y, 3z + 1 etc.

A polynomial of degree two is called a quadratic polynomial. e.g., 2x² + 3, 2x² + 3x - 4, 4x² etc.

Cubic Polynomials

A polynomial of degree three is called a quadratic polynomial. e.g., 2x³ + 3, 3x³ - 2x² + 3, 9x³ etc.

Root of the equation

A real number ‘a’ is a zero of a polynomial p(x) if p(a) = 0. In this case, a is also called a root of the equation p(x) = 0.

Remainder Theorem

If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

Factor Theorem

The linear polynomial x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if x – a is a factor of p(x), then p(a) = 0.

Factorization

Factors of polynomials in a small part i.e., called factorization.

Relationship between Zeroes and Coefficients of a Polynomial

In general form of quadratic polynomial, p(x) = ax² + bx + c,
Sum of its zeroes (α + β) = -b/a = (- coefficient of x)/(coefficient of x²)
Product of zeroes (αβ) = c/a = (constant term)/(coefficient of x²)

In general form of cubic polynomial, p(x) = ax³ + bx² + cx + d,
α + β + γ = -b/a
αβ + βγ + γα = c/a
αβγ = -d/a

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