📘 Polynomials Class 10 – Chapter 2
Class: 10 | Subject: Mathematics | Board: CBSE
🌟 Introduction to Polynomials
A polynomial is an algebraic expression consisting of variables, constants and non-negative integral powers of variables. These expressions form an important part of algebra and help us understand various mathematical relationships.
In this chapter, students will explore different types of polynomials and their degrees. Furthermore, they will learn about zeroes, their geometrical interpretation and the relationship between zeroes and coefficients.
The concepts covered in Polynomials Class 10 are useful for developing algebraic skills. Therefore, understanding the definitions, graphs and important formulas is essential for solving CBSE examination questions.
1️⃣ What is a Polynomial?
A polynomial in one variable x is an algebraic expression of the following form:
Here, a₀, a₁, a₂, ... are real numbers, and the powers of x are non-negative integers. Additionally, the coefficients must be real numbers.
Types of Polynomials Based on Degree
Polynomials can be classified according to their highest power of the variable. The following table explains the three common types:
| Type | General Form | Degree |
|---|---|---|
| Linear Polynomial | ax + b | 1 |
| Quadratic Polynomial | ax² + bx + c | 2 |
| Cubic Polynomial | ax³ + bx² + cx + d | 3 |
In these general forms, the coefficient of the highest power must be non-zero.
Consider the polynomial:
The highest power of x in this expression is 3. Therefore, its degree is 3, making it a cubic polynomial.
2️⃣ Degree of a Polynomial
The degree of a polynomial is the highest power of its variable with a non-zero coefficient. In other words, it determines whether the polynomial is linear, quadratic, cubic or of a higher degree.
p(x) = 5x² + 3x + 8
The highest power of x is 2. Hence, the degree of the polynomial is 2.
✏️ Example 2:p(x) = 7x⁴ - 3x² + 2
Here, the highest power is 4. Therefore, the degree is 4.
A non-zero constant polynomial has degree zero. However, the degree of the zero polynomial is not defined.
3️⃣ Zeroes of a Polynomial
A real number k is called a zero of a polynomial p(x) if the value of the polynomial becomes zero when x = k.
In simple words, zeroes are the values of the variable for which the polynomial becomes equal to zero.
How to Find the Zeroes of a Polynomial?
To find the zeroes, substitute different values of x into the given polynomial. The values that make the expression equal to zero are its zeroes.
Find the zeroes of:
Step 1: Put x = 2.
p(2) = 2² - 5(2) + 6
= 4 - 10 + 6 = 0
Therefore, 2 is a zero.
Step 2: Put x = 3.
p(3) = 3² - 5(3) + 6
= 9 - 15 + 6 = 0
Similarly, 3 is also a zero.
Hence, the zeroes of the polynomial are 2 and 3.
4️⃣ Geometrical Meaning of Zeroes of a Polynomial
The zeroes of a polynomial can also be understood through its graphical representation. For this purpose, we draw the graph of y = p(x) on the Cartesian plane.
The points where the graph intersects or touches the x-axis represent the zeroes of the polynomial. At these points, the value of y becomes zero.
The number of distinct real zeroes of a polynomial is equal to the number of distinct points where its graph intersects or touches the x-axis.
Number of Zeroes According to Graphs
The geometrical representation helps us understand how many real zeroes a polynomial can have.
- Linear Polynomial: Its graph is a straight line that intersects the x-axis at exactly one point.
- Quadratic Polynomial: Its graph is a parabola. Depending on its position, it can have zero, one or two distinct real zeroes.
- Cubic Polynomial: Its graph can have one, two or three distinct real zeroes.
A polynomial of degree n can have at most n distinct real zeroes.
For example, a quadratic polynomial can have at most two distinct real zeroes.
5️⃣ Relationship Between Zeroes and Coefficients
One of the most important concepts in Polynomials Class 10 is the relationship between zeroes and coefficients of a quadratic polynomial.
This relationship allows us to calculate the sum and product of zeroes directly from the coefficients. Consequently, we do not always need to solve the polynomial to find these values.
🔹 Quadratic Polynomial
Consider a standard quadratic polynomial:
Let α and β represent its two zeroes.
1. Sum of Zeroes
Thus, the sum of the zeroes is equal to the negative of the coefficient of x divided by the coefficient of x².
2. Product of Zeroes
Similarly, the product of the zeroes is equal to the constant term divided by the coefficient of x².
Find the sum and product of the zeroes of:
Comparing with ax² + bx + c:
a = 2, b = -7 and c = 3
Sum of zeroes:
= -b/a
= -(-7)/2 = 7/2
Product of zeroes:
= c/a
= 3/2
Therefore, the sum is 7/2 and the product is 3/2.
6️⃣ Formation of a Quadratic Polynomial
A quadratic polynomial can also be formed when the sum and product of its zeroes are given. For this purpose, we use the following formula.
Let S represent the sum of zeroes and P represent their product.
Form a quadratic polynomial whose zeroes have a sum of 5 and a product of 6.
Solution:
Given:
Sum of zeroes = 5
Product of zeroes = 6
Using the formula:
p(x) = x² - (Sum)x + Product
= x² - 5x + 6
Required polynomial = x² - 5x + 6
📝 Important Formulas of Polynomials Class 10
For quick revision, remember the following formulas and mathematical relationships:
| Concept | Formula |
|---|---|
| Zero of a Polynomial | p(k) = 0 |
| Maximum Number of Zeroes | Degree of Polynomial |
| Sum of Quadratic Zeroes | α + β = -b/a |
| Product of Quadratic Zeroes | αβ = c/a |
| Formation of Quadratic Polynomial | x² - Sx + P |
Always compare the given quadratic polynomial with ax² + bx + c before applying the formulas. This simple step helps prevent sign errors while calculating the sum and product of zeroes.
🎯 Quick Revision – Polynomials Class 10
Before attempting the exercises, revise these important points:
- A polynomial is an algebraic expression with non-negative integral powers of variables.
- The highest power of the variable with a non-zero coefficient determines its degree.
- A zero of a polynomial makes its value equal to zero.
- Graphically, the zeroes are represented by the x-coordinates of the points where the graph meets the x-axis.
- A polynomial of degree n can have at most n distinct real zeroes.
- The sum of the zeroes of ax² + bx + c is -b/a.
- The product of the zeroes of ax² + bx + c is c/a.
- A quadratic polynomial can be formed using its sum and product of zeroes.
❓ Frequently Asked Questions (FAQs)
Here are some commonly asked questions about Polynomials Class 10 CBSE Mathematics. These answers will help students revise important concepts and clear common doubts.
1. What is a polynomial in Class 10 Mathematics?
A polynomial is an algebraic expression consisting of variables, constants and non-negative integral powers of variables. For example, 3x² + 5x + 2 is a polynomial.
2. What is the degree of a polynomial?
The highest power of the variable with a non-zero coefficient is called the degree of a polynomial. For instance, the degree of 4x³ + 2x² - 5x + 1 is 3.
3. What are the zeroes of a polynomial?
The zeroes of a polynomial are the values of the variable for which the polynomial becomes zero. Mathematically, if p(k) = 0, then k is a zero of p(x).
4. How many zeroes can a quadratic polynomial have?
A quadratic polynomial can have at most two distinct real zeroes. Depending on its graph, it may have no real zero, one distinct real zero or two distinct real zeroes.
5. How many zeroes can a cubic polynomial have?
A cubic polynomial can have at most three distinct real zeroes. Therefore, it may have one, two or three distinct real zeroes.
6. What is the geometrical meaning of zeroes of a polynomial?
The zeroes of a polynomial represent the x-coordinates of the points where its graph intersects or touches the x-axis. At these points, the value of the polynomial is zero.
7. What is the relationship between zeroes and coefficients of a quadratic polynomial?
For a quadratic polynomial ax² + bx + c, with zeroes α and β, the sum of zeroes is -b/a and their product is c/a. These formulas help us calculate the required values directly from the coefficients.
8. How can we form a quadratic polynomial when its zeroes are given?
If α and β are the given zeroes, the required quadratic polynomial can be formed using x² - (α + β)x + αβ. Here, α + β represents the sum and αβ represents the product of the zeroes.
9. What is the maximum number of zeroes of a polynomial?
A polynomial of degree n can have at most n distinct real zeroes. For example, a linear polynomial has at most one, while a quadratic polynomial has at most two distinct real zeroes.
10. Is a constant polynomial a zero polynomial?
No. A non-zero constant polynomial has degree zero and no zeroes. However, the zero polynomial is a special case because its value is always zero and its degree is not defined.
11. Which formulas are important in Polynomials Class 10?
The most important formulas are the sum of zeroes (α + β = -b/a) and the product of zeroes (αβ = c/a) for a quadratic polynomial ax² + bx + c. Additionally, students should understand the geometrical interpretation of zeroes.
12. How should students prepare the Polynomials chapter for CBSE Class 10?
Students should first understand the definitions and geometrical meaning of zeroes. Next, they should practise questions based on the relationship between zeroes and coefficients. Finally, solving NCERT exercises, sample papers and additional worksheets will improve accuracy and confidence.
📚 Practice More with SkillShala
📖 Chapter 1 – Real Numbers
Before moving ahead with Polynomials, strengthen your understanding of the previous chapter, Real Numbers. SkillShala provides a dedicated worksheet to help Class 10 students practise important questions.
👉 Download Real Numbers Worksheet – Class 10
🎓 Explore More Learning Resources
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🏆 Conclusion
Polynomials is an important chapter in Class 10 Mathematics because it builds a strong foundation in algebra. Understanding zeroes, graphical representation and the relationship between zeroes and coefficients makes the chapter easier to solve.
Furthermore, regular practice of numerical problems and NCERT exercises will help students develop confidence. Therefore, revise the important formulas and practise different question types before your CBSE examination.
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