Polynomials Class 10 Notes CBSE Maths Chapter 2 (Free PDF Download)

• If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$, the degree of the polynomial \[\text{p}\left( \text{x} \right)\] is the largest power of $\text{x}$ in \[\text{p}\left( \text{x} \right)\].

• Types of Polynomials:

1. A linear polynomial is a polynomial with degree one.

2. A quadratic polynomial is a polynomial with degree two.

3. A cubic polynomial is a polynomial with degree three.

• Zeros of a Polynomial:

If \[\text{p}\left( \text{x} \right)\] is a polynomial in $\text{x}$ and $\text{k}$ is any real number, the value obtained by substituting $\text{k}$ for $\text{x}$ in \[\text{p}\left( \text{x} \right)\] is known as the value of \[\text{p}\left( \text{x} \right)\] when \[\text{x = k}\]and is denoted by \[\text{p}\left( \text{k} \right)\]. If \[\text{p}\left( \text{k} \right)\text{ = 0}\], a real number $\text{k}$ is said to be a zero of a polynomial \[\text{p}\left( \text{x} \right)\].

• The Geometrical Meaning of Polynomial Zeros:

• The equation \[\text{a}{{\text{x}}^{\text{2}}}\text{ + bx + c}\] can have three cases for the graphs 

1. Case (i): 

Here, the graph cuts \[\text{x-}\]axis at two distinct points \[\text{A}\]and \[\text{A }\!\!'\!\!\text{ }\].

2. Case (ii): Here, the graph cuts the \[\text{x-}\]axis at exactly one point.

3. Case (iii): Here, the graph is either completely above the \[\text{x-}\]axis or completely below the \[\text{x-}\]axis.

• If \[\text{ }\!\!\alpha\!\!\text{ }\] and \[\text{ }\!\!\beta\!\!\text{ }\] are the zeroes of the quadratic polynomial\[\text{p}\left( \text{x} \right)\text{ = a}{{\text{x}}^{\text{2 }}}\text{+ bx + c, a}\ne \text{0}\], then it is known that \[\text{x -  }\!\!\alpha\!\!\text{ }\] and \[\text{x -  }\!\!\beta\!\!\text{ }\] are the factors of \[\text{p}\left( \text{x} \right)\].

1. \[\text{ }\!\!A\!\!\text{  +  }\!\!\beta\!\!\text{  = - }\dfrac{\text{b}}{\text{a}}\]

2. \[\text{ }\!\!\alpha\!\!\text{  }\!\!\beta\!\!\text{  = }\dfrac{\text{c}}{\text{a}}\]

• Division Algorithm for Polynomials: 

• If \[\text{p}\left( \text{x} \right)\] and \[\text{g}\left( \text{x} \right)\] are any two polynomials with \[\text{g}\left( \text{x} \right)\ne 0\], then polynomials \[\text{q}\left( \text{x} \right)\] and \[\text{r}\left( \text{x} \right)\] can be found such that  \[\text{p}\left( \text{x} \right)\text{= g}\left( \text{x} \right)\text{  }\!\!\times\!\!\text{  q}\left( \text{x} \right)\text{ + r(x)}\], where \[\text{r}\left( \text{x} \right)=0\] or degree of \[\text{r}\left( \text{x} \right)<\] degree of \[\text{g}\left( \text{x} \right)\].

• This result is known as the Division Algorithm for polynomials.  

• An example would make it easier to understand. So, consider a cubic polynomial x³ - 3x² - x + 3.

• Assuming that one of its zeroes is $1$, it is clear that \[\text{x - 1}\] is a factor of x³ - 3x² - x + 3.

• So, x³ - 3x² - x + 3 can be divided by x -1 Taking out this factor, (x - 1)(x² - 2x - 3).

• Next, get the factors of x² - 2x - 3 by splitting the middle term. (x + 1)(x - 3).

x³−3x²−x+3=(x−1)(x+1)(x−3)

• So, all the three zeroes of the cubic polynomial are  $1, − 1, 3$.

Mastering Polynomials: Comprehensive Class 10 Notes for Polynomial - A Chapter Overview

Class 10 Maths Chapter 2 Notes is available in a PDF format on the main website of Vedantu. Students can avail of this opportunity to take a physical copy, which avoids internet issues. This pdf is also helpful to store for the future. It is also useful to refer during the time of entrance test, olympiads, and other examinations.