CBSE Class 10 Maths Revision Notes Chapter 2

Class 10 Mathematics Revision Notes for Polynomials of Chapter 2

The Class 10 Mathematics Chapter 2 Notes are prepared according to the latest CBSE syllabus covering all the important questions. Students can rely on these notes to prepare for the board examinations. Various topics such as factorisation, the relationship between the zeros and coefficient of polynomials, graphical representations of polynomial equations, polynomial expressions, and many more are discussed in detail. So let us start!

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Polynomial Class 10 Notes Polynomials – Chapter at a Glance

• Algebraic Expressions

It is an expression made of constants and variables with different mathematical operations. An algebraic expression can have any number of terms. The coefficient in each term can be a real number, but the exponents on the variables must be rational numbers. 

• Polynomials 

 Polynomials are algebraic expressions that can have exponents as rational numbers.

Example: Let us take an 5x3 + 3x + 1

In this 2x + 3√x is an algebraic expression but not a polynomial as the exponent on x is not a whole number. 

• Degree of polynomial

The highest exponent on the variable in a polynomial is known as the degree of the polynomial. 

Example: The degree of the polynomial x2 + 2x + 3 will be 2 because the highest power of x in the given expression is x2.

• Types of polynomials 

Number of terms 

Degree of polynomial 

• Types of polynomials based on the number of terms 

Monomial: A polynomial with a single term. 

Binomial: A polynomial with two different terms. 

Trinomial: A polynomial with three different terms. 

• Types of polynomials based on the degree

Linear polynomial: 

A polynomial of degree one is called the linear polynomial. 

Example: 2x + 1

Quadratic polynomial: 

A polynomial of degree two is known as a quadratic polynomial. 

Example: 3x2 + 5x +9 

Cubic polynomial: 

A polynomial with degree three is known as a cubic polynomial. 

Example: 2x3 + 5x2 + 6x + 15

Polynomials Class 10 Revision Notes Free PDF

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Revision Notes for Class 10 Chapter 2 Polynomials

Students can easily access Class 10 Chapter 2 Mathematics Notes on Extramarks to prepare for the board examination. Candidates can rely on the study material because it is created from the examination point of view. Candidates should review the available study materials for a better understanding before practising numerical and CBSE revision notes.

Geometrical Meaning of The Zeros of A Polynomial

• Number of zeros
• A linear polynomial will have one zero. 
• A quadratic polynomial will have at most two zeros. 
• A cubic polynomial expression will have at most three zeros. 
• The graphical representation of the zeros of a polynomial is as follows: 
• The first graph shows a linear polynomial representation. 
• The second graph shows two zeros or the quadratic polynomial. 
• The third graph shows three zeros or the cubic polynomial. 

Suggested to provide labelling

Relationship Between Zeros And Coefficients of a Polynomial

Let us take an example to understand the relationship between the zeros and coefficients of a polynomial. 

Let us consider p (x) = 2x2 – 8x + 6

Let’s split the middle term -8x as a sum of two terms.

We will write it as

2x2 – 8x + 6 = 2x2 – 6x – 2x + 6 = 2x ( x -3 ) -2 ( x – 3 )

      = ( 2x – 2 ) ( x – 3) = 2 (x – 1) (x – 3)

Now the value of p (x) = 2x2 – 8x + 6 is zero when x – 1 = 0 or x -3 = 0 that is, when x = 1 or x = 3. The zeros of the polynomial 2x2 – 8x + 6 are 1 and 3.  

Sum of the zeros will be 1 + 3 = 4 = – (-8)/2 = – coefficient of x/coefficient of x2 

Product of the zeros will be 1 X 3 = 3 = 6/2 = constant term/coefficient of x2 

Generally, if α and β are the zeros of a quadratic equation p (x) = ax2 + bx + c, where a is not equal to zero, then we get x – α and x – β are the factors of p(x). Hence, 

ax2 + bx + c = k(x – α) (x – β), where k is a constant 

                   = k[x2  – (α + β)x + α β]

                   = kx2  – k(α + β)x + k α β

Now comparing the coefficient of x2, x and constant term on both sides, we get                                      

a = k, b = – k(α + β) and c = kαβ

We get,           α + β = –b/a 

                          αβ = c/a

The sum of zeros = α + β = -b/a = – coefficient of x/coefficient of x2 

Product of zeros =  αβ = c/a = constant term/coefficient of x2

Division Algorithm For Polynomials

Let us take the following example to understand the division algorithm for polynomials.

Example: 3x3 + x2 + 2x + 5 by 1 + 2x + x2 

• Step 1: To get the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor. The answer will be 3x. Now carry out the division process. You will get -5x2 – x + 5.
• Step 2: To get the second term of the quotient, divide the highest degree term of the new dividend ( – 5x2 ) by the highest degree term of the divisor ( x2). The answer will be -5. Now carry out the division process again. 
• Step 3: 9x + 10 remains after the division. Now the degree of 9x + 10 is less than the degree of the divisor, so x2 + 2x + 1. So, we cannot divide it any further.

The quotient will be 3x – 5, and the remainder will be 9x + 1. Also, 

(x2 + 2x + 1) X (3x – 5) + (9x + 10) = 3x3 + 6x2 + 3x – 5x2 – 10x – 5 + 9x + 10

      = 3x3 + x2 + 2x + 5

We observed that 

Dividend = Divisor X Quotient + Remainder 

If p (x) and g (x) are two polynomials with g (x) not equal to 0, then we find that 

p (x) = g (x) X q (x) + r(x)

Where r(x) = 0 or degree of r(x) < degree of g(x).

Summary

• Degrees 1, 2, and 3 of polynomials are referred to as linear, quadratic, and cubic, respectively.
• With real coefficients, a quadratic polynomial in x is in the form of ax2 + bx + c where a, b and c are real numbers with a not equal to zero. 
• The zeros of the polynomial p(x) are the x coordinates of the points where the graph of y = p(x) intersects at x-axis. 
• A quadratic polynomial can have maximum 2 zeros and a cubic polynomial can have 3 zeros at most. 
• If α and β are zeros of the quadratic polynomial ax2 + bx + c, then 

α +β = -b/a, αβ = c/a

• If  α, β, γ are zeros of the cubic polynomial ax3 + bx2 + cx + d, then 

α +β + γ = -b/a,

αβ+ βγ+ γα = c/a,

and      αβγ = -d/a

• The division algorithm states that for any given polynomial p(x) and any non-zero polynomial g(x), there are polynomials q(x) and r(x) such that 

p(x) = g(x) q(x) + r(x),

Where r(x) = 0 or degree r(x) < degree g(x).