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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 xaxis.
 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 nonzero 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).
FAQs (Frequently Asked Questions)
1. Mention the various algebraic identities related to polynomials.
The algebraic identities are as follows:
 (a+b)2=a2+2ab+b2
 (a−b)2=a2−2ab+b2
 (x+a)(x+b)=x2+(a+b)x+ab
 a2−b2=(a+b)(a−b)
 a3−b3=(a−b)(a2+ab+b2)
 a3+b3=(a+b)(a2−ab+b2)
 (a+b)3=a3+3a2b+3ab2+b3
 (a−b)3=a3−3a2b+3ab2−b3
2. How can I access the sample papers for practice?
Students can access the CBSE sample papers and CBSE past years’ question papers on Extramarks to get clarity about the types of questions asked in the examination. They should also practise all the numerical and formulas to score well in the examinations.
3. How can I ace the board examinations?
To ace the board examinations, candidates in Class 10 should thoroughly practise all of the CBSE extra questions, important questions and formulas are given in the CBSE syllabus. The regular practice of numerical will help students improve their problemsolving skills.