CBSE Class 10 Maths Notes Polynomial:Download PDF Here
Polynomials for class 10 concepts are given in detail. Go through the below article to learn the concept of polynomials such as expressions, degrees, types, graphical representation and so on.
Algebraic Expressions
An algebraic expression is an expressionÂ made up of variables and constants along with mathematical operators.
An algebraic expression is a sum of terms, which are considered to be building blocks forÂ expressions.
A term is a product of variables andÂ constants. A term can be an algebraic expression in itself.
Examples of a termÂ – 3 which is just a constant.
– 2x, which is the product of constant ‘2’ and the variable ‘x’
– 4xy, which is the product of the constant ‘4’ and the variables ‘x’ and ‘y’.
– 5x2y, which is the product of 5, x, x and y.
The constant in each term is referred to as theÂ coefficient.
Example of an algebraic expression – 3x2y+4xy+5x+6 – which is the sum of four terms – 3x2y,Â 4xy,Â 5x and 6
An algebraic expression can have any number of terms. The coefficient in each term can be any real number. There can be any number of variables in an algebraicÂ expression. The exponent on the variables, however, must be rational numbers.
Polynomial
An algebraic expression can have exponents that areÂ rational numbers. However, a polynomial is an algebraic expression in which the exponent on any variable is a whole number.
5x3+3x+1 is an example of a polynomial. It is an algebraic expression as well
2x+3âˆšxÂ is an algebraic expression, but not a polynomial. – since the exponent on x is 1/2 which is not a whole number.
Degree of a Polynomial
For a polynomial in one variableÂ – the highest exponent on the variable in a polynomial is the degree of the polynomial.
Example:Â The degree of the polynomial x2+2x+3 is 2, as the highest power of x in the given expression is x2.
TYPES OF POLYNOMIALS
Polynomials can be classified based on
a) Number of terms
b) Degree of the polynomial.
Types of polynomials based on the number of terms
a) Monomial – A polynomial with just one term. Example – 2x, 6x2, 9xy
b)Â Binomial – A polynomial with two terms. Example – 4x2+x, 5x+4
a) Trinomial – A polynomial with three terms. Example – x2+3x+4
Types of Polynomials based on Degree
Linear Polynomial
A polynomial whose degree is one is called a linear polynomial.
For example, 2x+1 is a linear polynomial.
Quadratic Polynomial
A polynomial of degree two is called a quadratic polynomial.
For example, 3x2+8x+5 is a quadratic polynomial.
Cubic Polynomial
A polynomial of degree three is called a cubic polynomial.
For example, 2x3+5x2+9x+15 is a cubic polynomial.
Graphical Representations
Representing Equations on a Graph
Any equation can be represented as a graph on the Cartesian plane, where each point on the graph represents the x and y coordinates of the point that satisfies the equation. An equation can be seen as a constraint placed on the x and y coordinates of a point, and any point that satisfies that constraint will lie on the curve
For example, the equation y = x, on a graph, will be a straight line that joins all the points which have their x coordinate equal to their y coordinate. Example – (1,1), (2,2) and so on.
Visualization of a Polynomial
Geometrical Representation of a Linear Polynomial
The graph of a linear polynomial isÂ a straight line. It cuts the Xaxis at exactly one point.
Geometrical Representation of a Quadratic Polynomial

 The graph of a quadratic polynomial is a parabola.
 It looks like a U which either opens upwards or opens downwards depending on the value of a in ax2+bx+c.
 If a is positive then parabola opens upwards and if a is negative then it opens downwards.
 It can cut the xaxis at 0, 1 or two points.
Graph of the polynomial xn
For a polynomial of the form y=xn where n is a whole number:
 as n increases, the graph becomes steeper or draws closer to the Yaxis.
 If n is odd, the graph lies in the first and third quadrants
 If n is even, the graph lies in the first and second quadrants.
 The graph of y=âˆ’xn is the reflection of the graph ofÂ y=xn on the xaxis
Zeroes of a Polynomial
A zero of a polynomial p(x) is the value of x for which the value ofÂ p(x) is 0. If k is a zero of p(x), then p(k)=0.
For example, consider a polynomial p(x)=x2âˆ’3x+2.
When x=1, the value of p(x) will be equal to
p(1)=12âˆ’3Ã—1+2
=1âˆ’3+2
=0
Since p(x)=0 at x=1, we say that 1 is a zero of the polynomial x2âˆ’3x+2
Geometrical Meaning of Zeros of a Polynomial
Geometrically, zeros of a polynomial are the points where its graph cuts the xaxis.
Here A, B and C correspond to the zeros of the polynomial represented by the graphs.
Number of Zeros
In general a polynomial of degree n has at most n zeros.
 A linear polynomial has one zero,
 A quadratic polynomial has at most two zeros.
 A cubic polynomial has at most 3 zeros.
Factorization of Polynomials
The factorisation of Quadratic Polynomials
Quadratic polynomials can be factorized by splitting the middle term.
For example, consider the polynomial 2x2âˆ’5x+3
SplittingÂ the middle term.
The middle term in the polynomialÂ 2x2âˆ’5x+3 is 5. This must be expressed as a sum of two terms such that the product of their coefficients is equal to the product of 2 and 3 (coefficient of x2 and the constant term)
âˆ’5 can be expressed as (âˆ’2)+(âˆ’3), as âˆ’2Ã—âˆ’3=6=2Ã—3
Thus,Â 2x2âˆ’5x+3=2x2âˆ’2xâˆ’3x+3
Now, identify the common factors in individual groups
2x2âˆ’2xâˆ’3x+3=2x(xâˆ’1)âˆ’3(xâˆ’1)
Taking (xâˆ’1) as the common factor, this can be expressed as
2x(xâˆ’1)âˆ’3(xâˆ’1)=(xâˆ’1)(2xâˆ’3)
Relationship between Zeroes and Coefficients
Relationship between Zeroes and Coefficients of a Polynomial
If Î± and Î² are the roots of Â a quadratic polynomial ax2+bx+c, then,
Î± +Â Î² = b/a
Sum of zeroes = = coefficient of x /coefficient ofÂ x2
Î±Î² = c/a
product of zeroes = = constant term / coefficient ofÂ x2
If Î±,Î²Â andÂ Î³Â are the roots of a cubic polynomial ax3+bx2+cx+d, then
Î±+Î²+Î³ = b/aÂ
Î±Î² +Î²Î³ +Î³Î± = ca
Î±Î²Î³ = da
Division Algorithm
Division Algorithm for a Polynomial
To divide one polynomial by another, follow the steps given below.
Step 1: arrange the terms of the dividend and the divisor in the decreasing order of their degrees.
Step 2: To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisorÂ Then carry out the division process.
Step 3: The remainder from the previous division becomes the dividend for the next step. Repeat this process until the degree of theÂ remainder is less than the degree of the divisor.
Algebraic Identities
1. (a+b)2=a2+2ab+b2
2. (aâˆ’b)2=a2âˆ’2ab+b2
3. (x+a)(x+b)=x2+(a+b)x+ab
4. a2âˆ’b2=(a+b)(aâˆ’b)
5. a3âˆ’b3=(aâˆ’b)(a2+ab+b2)
6. a3+b3=(a+b)(a2âˆ’ab+b2)
7. (a+b)3=a3+3a2b+3ab2+b3
8. (aâˆ’b)3=a3âˆ’3a2b+3ab2âˆ’b3