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A polynomial is an algebraic expression that contains a variable and a degree. In polynomial theory, a variable is also called an indeterminate, and the degree is known as the highest power of the variable. Polynomial expressions are made by arithmetic operations like addition, subtraction, and multiplication. These arithmetic operations are used to define the relationship between variables. Polynomials are used in many areas, like science, economics, mathematics, engineering, etc. In these areas, polynomials are considered as the fundamental tool for analysis and solving problems.
These notes contain all the topics, subtopics, short tricks, formulae, and all the required important key points. These notes cover the basic definition of polynomials, algebraic expressions, degree of polynomials, and types of polynomials and their graphical representation. CBSE Class 10 chapter Polynomials also includes the geometrical representation of the zeroes of polynomials. Students must practice all the topics of polynomials and their examples from the NCERT Exemplar Solutions for Class 10 Maths Chapter 2 Polynomials. Students must practice questions and check their solutions using the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials.
Polynomials: It is an algebraic expression that can have an exponent as a real number or a whole number. A polynomial represented by
Algebraic Expressions: An algebraic expression is an equation with variables and constants that is made up of arithmetic operators.
Polynomials can be divided into two categories based on the following characteristics:
1. Number of terms
2. Degree of polynomials
A polynomial expression can have one or more terms, and according to the number of terms, polynomials are as follows:
1. Monomials: A polynomial that has only one term is called a monomial.
Example:
2. Binomials: A polynomial that has two unlike terms is called a binomial.
Example:
3. Trinomials: A polynomial that has three unlike terms is called a trinomial.
Example:
1. Zero Polynomials: A polynomial that has 0 as a degree is called a zero or constant polynomial.
Example:
2. Linear Polynomial: A polynomial that has 1 as a degree is called a linear polynomial.
Example:
3. Quadratic Polynomial: A polynomial that has 2 as a degree is called a quadratic polynomial.
Example:
4. Cubic Polynomial: A polynomial that has 3 as a degree is called a cubic polynomial.
Example:
The value of the polynomials that make the polynomials equal to zero.
Example: For the polynomial
The representation of different types of polynomials is as follows.
When the x-coordinate of the point of intersection is at the x-axis of the graph, then this is called geometrically the zeroes of a polynomial.
The general form of the linear polynomial is y = ax + b, where a must not be equal to 0. For example, the equation is y = 3x + 1 for x = 2, and -2, the line of the graph passing through the points (2, 7) and (-2, -5)
x | 2 | -2 |
y = 3x + 1 | 7 | -5 |
The graph is shown below:
The general form of the quadratic polynomials is ax² + bx + c, where a is not equal to zero. The graph for a quadratic equation looks like a symbol
The geometrical representation of quadratic polynomials when a > 0 and a < 0 is as follows:
According to the shape of the graph, there are three possible cases:
Case 1: When the graph cuts the x-axis at two distinct points, P and P'. Here, P and P' are the zeros of the quadratic equation ax² + bx + c.
Case 2: When the graph cuts the x-axis at exactly one point, i.e., at two coincident
points. Therefore, the two points P and P' of Figure 1 of case 1 coincide here to become one point P. So, the x-coordinate of P is the only zero for the quadratic polynomial ax² + bx + c.
Case 3: When the graph is either completely above the x-axis or completely below
the x-axis, then it does not cut the x-axis at any point. So, the quadratic polynomial ax2
+ bx + c has no zero in this case.
For quadratic Polynomials: Let α and β be the roots of the quadratic equation ax2
+ bx + c, then,
Sum of the zeroes (α + β) =
Product of the zeroes (αβ) =
Example: x2 + 7x + 10
Sum of the zeroes (α + β) =
Product of the zeroes (αβ) =
For Cubic Polynomials: Let α, β and γ be the roots of a cubic polynomial ax³ + bx² + cx + d; then
α + β + γ =
αβ + βγ + γα =
αβγ =
If P(x), D(x), Q(x) and R(x) are the three polynomials where,
P(x) = Polynomial
D(x) = Divisor polynomial
Q(x) = Quotient polynomial
R(x) = Remainder polynomial
Therefore, the formula for the division algorithm is as follows:
P(x) = D(x) × Q(x) + R(x)
Where D(x) ≠ 0, and R(x) = 0 or the degree of R(x) < the degree of D(x).
Example: Divide the cubic polynomial 4x3 + 3x2 + 3x + 2 by the quadratic polynomial x2 + 2x + 1
The division is shown in the figure:
Proof Using Divisor Algorithm:
P(x) = D(x) × Q(x) + R(x)
P(x) = 4x3 + 3x2 + 3x + 2
D(x) = x2 + 2x + 1
Q(x) = 4x - 5
R(x) = 9x + 7
Now, putting the values in the formula, we get:
4x3 + 3x2 + 3x + 2 = (x2 + 2x + 1) × (4x - 5) + 9x + 7
4x3 + 3x2 + 3x + 2 = 4x3 + 8x2 + 4x - 5x2 - 10x - 5 + 9x + 7
4x3 + 3x2 + 3x + 2 = 4x3 + 3x2 + 3x + 2
L.H.S. = R.H.S.
Students must download the notes below for each chapter to ace the topics.
NCERT Class 10 Solutions for Maths And Science
Students must check the NCERT solutions for Class 10 Maths and Science given below:
Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:
Read More: Syllabus and Books
A polynomial is an algebraic expression that can have an exponent as a real number or a whole number. A polynomial represented by
Polynomials can be divided into two categories based on the number of terms and based on degree. Their subcategories are monomials, binomials, trinomials, zero polynomials, quadratic polynomials, and cubic polynomials.
Linear polynomials have a degree of 1
(example: 2x + 3)
Quadratic polynomials have a degree of 2
(Example: 2x² + 5 + 2)
Cubic polynomials have a degree of 3
(Example: 3x³ + 2x² + x + 2)
Students can determine zeros of polynomials by setting the equation equal to zero and determining the factors of the equation, and solve them or by applying the formula of sum of the zeroes (α + β) =
For quadratic equation the sum of the zeroes (α + β) =
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