CBSE Class 10 Maths Notes Polynomial

Introduction:
Mathematics is a subject that is both fascinating and challenging. In Class 10, one of the fundamental topics in mathematics is polynomials. Polynomials play a crucial role in various branches of mathematics, including algebra and calculus. In this article, we will explore the concept of polynomials, their algebraic expressions, and different types of polynomials. We will also discuss the degree of a polynomial and its significance. So, let's dive into the world of polynomials!

Algebraic expressions:
Before delving into polynomials, let's first understand algebraic expressions. An algebraic expression is a combination of variables, constants, and arithmetic operations like addition (+), subtraction (-), multiplication (×), and division (÷). For example, the expression 2x + 3y - 5 is an algebraic expression, where 'x' and 'y' are variables and 2, 3, and 5 are constants.

Polynomial:
A polynomial is an algebraic expression that consists of one or more terms. Each term in a polynomial has a variable raised to a non-negative integer power, multiplied by a coefficient. The general form of a polynomial is given by:

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀

Here, P(x) represents the polynomial function, 'a' represents the coefficients, 'x' represents the variable, and 'n' represents the highest power of 'x' in the polynomial. The coefficients can be any real numbers.

Degree of a polynomial:
The degree of a polynomial is the highest power of the variable 'x' in the polynomial expression. It helps us understand the behavior and properties of the polynomial. To find the degree of a polynomial, we examine the exponents of the terms and determine the term with the highest exponent. For example, if we have the polynomial 3x³ + 2x² - 5x + 1, the degree of the polynomial is 3 because the term with the highest power of 'x' is x³.

Types of polynomials:
Polynomials can be classified into different types based on their degree and the number of terms they contain. Let's explore some common types of polynomials:

1. Constant polynomial:
A constant polynomial is a polynomial of degree zero. It has only one term that consists of a constant. For example, P(x) = 5 is a constant polynomial. Regardless of the value of 'x', the polynomial will always be equal to 5.

2. Linear polynomial:
A linear polynomial is a polynomial of degree one. It has one term involving the variable raised to the power of one. For example, P(x) = 3x + 2 is a linear polynomial. The highest power of 'x' in this polynomial is 1.

3. Quadratic polynomial:
A quadratic polynomial is a polynomial of degree two. It has one term involving the variable raised to the power of two. For example, P(x) = 2x² - 3x + 1 is a quadratic polynomial. The highest power of 'x' in this polynomial is 2.

4. Cubic polynomial:
A cubic polynomial is a polynomial of degree three. It has one term involving the variable raised to the power of three. For example, P(x) = x³ + 4x² - 2x - 5 is a cubic polynomial. The highest power of 'x' in this polynomial is 3.

5. Higher-degree polynomials:
Polynomials can have degrees greater than three as well. These are often referred to as higher

-degree polynomials. For example, P(x) = 4x⁵ - 2x³ + x² + 3x - 1 is a polynomial of degree five.

Types of polynomials based on the number of terms:
Polynomials can also be classified based on the number of terms they contain. Let's explore some common types:

1. Monomial:
A monomial is a polynomial with only one term. It can be written in the form P(x) = aₙxⁿ, where 'a' is the coefficient and 'n' is a non-negative integer. For example, P(x) = 5x² is a monomial.

2. Binomial:
A binomial is a polynomial with two terms. It can be written in the form P(x) = aₙxⁿ + bₘxᵐ, where 'a' and 'b' are coefficients, and 'n' and 'm' are non-negative integers. For example, P(x) = 3x² - 2x is a binomial.

3. Trinomial:
A trinomial is a polynomial with three terms. It can be written in the form P(x) = aₙxⁿ + bₘxᵐ + cₚxᵖ, where 'a', 'b', and 'c' are coefficients, and 'n', 'm', and 'p' are non-negative integers. For example, P(x) = 2x³ - x² + 5x is a trinomial.

Types of Polynomials Based on Degree:
Apart from the classification based on the number of terms, polynomials can also be classified based on their degree. Let's explore these types:

1. Zero polynomial:
A zero polynomial is a polynomial in which all the coefficients are zero. It is represented by P(x) = 0. The degree of a zero polynomial is not defined since there are no non-zero terms.

2. Constant polynomial:
A constant polynomial is a polynomial of degree zero. It has only one term consisting of a constant. For example, P(x) = 7 is a constant polynomial.

3. Linear polynomial:
A linear polynomial is a polynomial of degree one. It has one term involving the variable raised to the power of one. For example, P(x) = 3x + 2 is a linear polynomial.

4. Quadratic polynomial:
A quadratic polynomial is a polynomial of degree two. It has one term involving the variable raised to the power of two. For example, P(x) = 2x² - 3x + 1 is a quadratic polynomial.

5. Cubic polynomial:
A cubic polynomial is a polynomial of degree three. It has one term involving the variable raised to the power of three. For example, P(x) = x³ + 4x² - 2x - 5 is a cubic polynomial.

Geometrical representations:
Polynomials can also be represented geometrically using graphs. The graph of a polynomial is a visual representation of the polynomial function. The x-axis represents the variable 'x', and the y-axis represents the corresponding values of the polynomial function. The shape of the graph depends on the degree and coefficients of the polynomial.

Zeroes of a polynomial:
The zeroes of a polynomial are the values of 'x' for which the polynomial function becomes zero. In other words, if P(x) is a polynomial, then any value of 'x' that satisfies P(x) = 0 is called a zero or root of the polynomial. The zeroes of a polynomial can be real or complex numbers.

Geometrical meaning of zeros of a polynomial:
The zeroes of a polynomial have geometrical significance when represented on a graph. The zeroes of a polynomial are the points where the polynomial function intersects the x-axis. These points are also known as the x-intercepts or roots of the polynomial. The graph of a polynomial will touch or cross the x-axis at the

locations corresponding to its zeroes.

Number of zeros:
The number of zeroes of a polynomial can be determined by its degree. For a polynomial of degree 'n', there can be at most 'n' zeroes. However, it is important to note that a polynomial can have fewer than 'n' zeroes, including repeated zeroes. The Fundamental Theorem of Algebra states that a polynomial of degree 'n' will have exactly 'n' zeroes when each zero is counted according to its multiplicity.

Factorisation of polynomials:
Factorisation is a crucial concept in mathematics that allows us to express a polynomial as a product of its factors. It helps us simplify and solve polynomial equations. The factorisation of a polynomial involves breaking it down into simpler terms, which can then be easily manipulated or solved. The factorisation process is based on identifying common factors and using algebraic techniques.

Relationship between zeroes and coefficients of a polynomial:
The zeroes of a polynomial and its coefficients are closely related. If 'a' is a zero of a polynomial function P(x), then (x - a) is a factor of the polynomial. Similarly, if (x - a) is a factor of the polynomial, then 'a' is a zero of the polynomial. This relationship between zeroes and factors helps us factorise polynomials and find their roots.

Division algorithm:
The division algorithm is a method used to divide polynomials. It allows us to divide a polynomial dividend by a polynomial divisor and obtain a quotient and remainder. The division algorithm helps in simplifying expressions, finding factors, and solving polynomial equations. It involves a step-by-step process of dividing the terms of the dividend by the terms of the divisor and obtaining the quotient and remainder.

Algebraic identities:
Algebraic identities are important formulas that establish relationships between different mathematical expressions. These identities help simplify expressions, solve equations, and manipulate algebraic equations efficiently. Some common algebraic identities involving polynomials include the square of a binomial, the cube of a binomial, and the difference of squares.

Polynomials for class 10 examples:
Now, let's address the five commonly made mistakes and provide detailed explanations for each concept. Here are the concepts along with example questions and step-by-step solutions:

1. Concept: factoring a quadratic polynomial
  Mistake: Incorrectly applying the factorization method.
  Example Question: Factorize the polynomial P(x) = x² + 5x + 6.
  Solution: To factorize the quadratic polynomial, we need to find two numbers whose sum is 5 and product is 6. The factors are 2 and 3. Thus, P(x) can be factored as (x + 2)(x + 3).

2. Concept: finding zeroes of a polynomial
  Mistake: Incorrectly identifying the zeroes of a polynomial.
  Example Question: Find the zeroes of the polynomial P(x) = 2x² - 8x + 6.
  Solution: To find the zeroes, we set P(x) equal to zero and solve the equation 2x² - 8x + 6 = 0. By factoring or using the quadratic formula, we find that the zeroes are x = 1 and x = 3.

3. Concept: applying algebraic identities
  Mistake: Incorrectly applying the algebraic identities.
  Example Question: Simplify the expression (2x + 3)² - (2x - 3)².
  Solution: We can use the identity (a + b)² - (a - b)² = 4ab to simplify the expression. Applying the identity, we get 4(2x + 3)(2x - 3), which further simplifies to 16x² - 9.

4. Concept: dividing polynomials
  Mistake: Making errors while performing polynomial division.
  Example Question: Divide the polynomial P(x) = 3x³ - 2x² + 5x - 1 by the polynomial Q(x) = x - 2.
  Solution: Using long division or synthetic division, we divide P(x) by Q(x).

The quotient is 3x² + 4x + 13, and the remainder is 25. Therefore, the division result is (3x² + 4x + 13) with a remainder of 25.

5. Concept: solving equations using polynomials
  Mistake: Incorrectly solving polynomial equations.
  Example Question: Solve the equation 2x³ - 5x² + 3x = 0.
  Solution: To solve the equation, we factor out the common term 'x' and obtain x(2x² - 5x + 3) = 0. Further factoring gives x(x - 1)(2x - 3) = 0. Therefore, the solutions are x = 0, x = 1, and x = 3/2.

By addressing these commonly made mistakes and providing detailed explanations with step-by-step solutions, students can gain a better understanding of polynomial concepts and improve their problem-solving skills.

Aha

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