NCERT Solutions for Class 10 Maths Chapter 2
Polynomials is the second chapter in the NCERT Maths textbook for Class 10. It introduces algebraic expressions known as polynomials and their various properties.
A Deep Dive into Polynomials: Building Blocks of Algebra
Moving beyond simple algebraic expressions, the chapter on Polynomials in Class 10 lays a crucial foundation for higher mathematics. It’s not just about solving equations; it’s about understanding the behavior, structure, and inherent relationships within these expressions. Think of it as learning the grammar of the language of algebra.
What Exactly is a Polynomial?
At its heart, a polynomial is a special kind of algebraic expression. But it has a strict rule: the exponents (or powers) of the variable must be whole numbers (non-negative integers). You can have constants (like 5, -2), variables (like *x*, *y*), and their products, but you cannot have variables in the denominator or under a radical sign.
For example:
- 4x³ – 3x² + 7 is a polynomial.
- √x + 5 or 1/(x+2) are not polynomials.
The highest power of the variable in a polynomial is called its degree. This isn’t just a label; it’s one of the most important characteristics, as it dictates the polynomial’s shape and properties.
A Family of Polynomials: Classified by Degree
Polynomials are often named based on their degree, which helps us instantly understand their complexity:
- Linear Polynomial: A polynomial of degree 1. Its graph is always a straight line. It has the general form ax + b, where a ≠ 0. For example, 2x – 5. A linear polynomial has exactly one zero.
- Quadratic Polynomial: A polynomial of degree 2. Its graph is a beautiful, U-shaped curve called a parabola. Its general form is ax² + bx + c, where a ≠ 0. Examples include x² – 5x + 6 or 2x² + 3. A quadratic polynomial can have two zeroes, or sometimes one repeated zero.
- Cubic Polynomial: A polynomial of degree 3. Its graph can have more twists and turns. Its general form is ax³ + bx² + cx + d, where a ≠ 0. An example is x³ – 4x² + x + 6. A cubic polynomial can have up to three zeroes.
The Heart of the Matter: Zeroes of a Polynomial
The zero of a polynomial, p(x), is a value of the variable ‘x’ for which the polynomial’s value becomes zero. In simpler terms, if p(α) = 0, then α is a zero of the polynomial.
Finding the zero is often the first step in solving real-world problems modelled by polynomials. Geometrically, the zeroes of a polynomial are the x-coordinates of the points where its graph intersects the x-axis.
The Beautiful Connection: Zeroes and Coefficients
This is where the chapter reveals a powerful and elegant link. We don’t always need to solve the polynomial to find its zeroes; we can find their sum and product by simply looking at its coefficients.
For a Quadratic Polynomial: If α and β are the zeroes of ax² + bx + c, then:
- Sum of zeroes (α + β) = -b/a
- Product of zeroes (αβ) = c/a
This relationship is a game-changer. If you’re given the polynomial x² – 5x + 6, you can instantly say that the sum of its zeroes is 5 and their product is 6, without ever solving x² – 5x + 6 = 0.
For a Cubic Polynomial: If α, β, and γ are the zeroes of ax³ + bx² + cx + d, then:
- α + β + γ = -b/a
- αβ + βγ + γα = c/a
- αβγ = -d/a
The Division Algorithm: A Formal Long Division
Just like we divide numbers, we can divide polynomials. The Division Algorithm provides a structured way to do this. It states that for any two polynomials p(x) (the Dividend) and g(x) (the Divisor), we can find polynomials q(x) (the Quotient) and r(x) (the Remainder) such that:
Dividend = (Divisor × Quotient) + Remainder
Or, p(x) = g(x) × q(x) + r(x)
The key rule here is that the degree of the remainder r(x) is always less than the degree of the divisor g(x). This process is vital for simplifying expressions and is a precursor to more advanced concepts like factorization.
Why Does This All Matter?
Understanding polynomials is not an academic exercise. The parabolic path of a basketball, the optimization of profit in a business, and the modelling of population growth can all be described using polynomials. By learning to find their zeroes, understand their graphs, and manipulate them through division, we acquire a fundamental toolkit for analysing and interpreting the world through a mathematical lens. This chapter equips you not just for your board exams, but for the logical and analytical thinking required in many future fields.
Exercise 2.1
1. The graphs of y = p(x) are given below for some polynomials p(x). Find the number of zeroes of p(x) in each case.
Ans :
Exercise 2.2
1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and their coefficients:
(i) x2 – 2x – 8
(ii) 4s2 – 4s + 1
(iii) 6x2 – 3 – 7x
(iv) 4u2 + 8u
(v) t2 – 15
(vi) 3x2 – x – 4
Ans :
NCERT Solutions for Class 10 Maths Chapter 2
FAQs
What is Chapter 2 of Class 10 Maths about?
Chapter 2, Polynomials, covers types of polynomials, zeroes, and their relationship with coefficients, along with examples and exercises.
How do NCERT Solutions help in Chapter 2 Polynomials?
They provide clear, step-by-step answers to all textbook questions, helping students understand formulas and solve problems easily.
What are the main topics covered in Polynomials Class 10?
Main topics include degree of a polynomial, zeroes of a polynomial, relationship between zeroes and coefficients, and division algorithm.
Are NCERT Solutions for Class 10 Maths Chapter 2 enough for board exams?
Yes, NCERT Solutions are sufficient for board exams as they follow the CBSE syllabus and cover all important concepts.
Where can I get free NCERT Solutions for Class 10 Maths Chapter 2?
You can access free NCERT Solutions for Class 10 Maths Chapter 2 – Polynomials – on reliable educational websites and learning platforms.
