What are the different types of polynomials in Class 10 Maths?
The concept of polynomials for class 10 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding polynomials is essential for success in algebra and forms the base for higher-level Maths topics, competitive exams, and practical applications.
What Is Polynomial for Class 10?
A polynomial for class 10 is an algebraic expression made up of variables (like x), constants, and coefficients combined using addition, subtraction, and multiplication, but with non-negative integer exponents only. You’ll find this concept applied in areas such as algebraic expressions, quadratic equations, and even graph plotting. For example, \( 2x^2 + 3x - 4 \) is a quadratic polynomial, and \( 4x + 7 \) is a linear polynomial.
Types of Polynomials for Class 10
In class 10, polynomials are mainly classified based on the number of terms and the degree (highest power of the variable).
| Type | Structure | Example |
|---|---|---|
| Monomial | 1 term | 7x, 4ab |
| Binomial | 2 terms | x + 5, 3x^2 − 2x |
| Trinomial | 3 terms | x^2 + 2x + 1 |
| Zero Polynomial | All coefficients zero | 0 |
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression. For example, in \( 5x^3 + 4x^2 - 2 \), the degree is 3. Degrees are used to classify polynomials as:
- Linear (degree 1): \( 7x - 5 \)
- Quadratic (degree 2): \( x^2 + x - 6 \)
- Cubic (degree 3): \( 2x^3 - x + 4 \)
Key Formulas for Polynomial for Class 10
Here’s the standard general form of a polynomial: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)
Some important formulas for class 10 polynomials include:
- Addition: Add corresponding terms.
- Subtraction: Subtract like terms.
- Multiplication: Multiply each term in one polynomial by every term in the other.
- Factor Theorem: If \( p(a) = 0 \), then \( (x−a) \) is a factor of \( p(x) \).
- Remainder Theorem: The remainder of \( p(x) \) divided by \( (x−a) \) is \( p(a) \).
Step-by-Step Example: Find the Degree
Let’s consider the expression \( 7x - 5 \):
1. The first term is \( 7x \), exponent is 1.2. The second term is \( -5 \), exponent is 0 (constant term).
3. The highest exponent is 1.
4. Therefore, the degree of \( 7x - 5 \) is 1.
Solved Question: Identify the Polynomial
Which of the following is a binomial?
- 8p + p
- 7p2 + 8q + 9r
- 3p × 4q × 2r
- 11p2 + 11q2
Answer:
1. \( 8p + p = 9p \) → Monomial2. \( 7p^2 + 8q + 9r \) → Trinomial
3. \( 3p × 4q × 2r = 24pqr \) → Monomial
4. \( 11p^2 + 11q^2 \) → Binomial
Hence, the answer is 11p2 + 11q2.
Speed Trick or Vedic Shortcut
When factorising quadratic polynomials, remember this shortcut:
For \( ax^2 + bx + c = 0 \), if the product \( ac \) can be split into two numbers whose sum is \( b \), you can quickly write the factors. Example: \( x^2 + 5x + 6 \), since 2 and 3 multiply to 6 and add up to 5, factors are \( (x+2)(x+3) \).
Common Mistakes in Polynomials
- Using negative exponents (these are not allowed in polynomials).
- Confusing trinomial with cubic polynomials (trinomial: three terms, cubic: degree 3).
- Missing out constant term when calculating the degree.
Relation to Other Maths Concepts
Mastering polynomials for class 10 helps you progress to quadratic equations and also makes it easier to understand polynomial equations and factorisation. Concepts such as algebraic expressions and factoring rely on this foundation.
Try These Yourself
- Classify \( x^3 + 2x^2 - 5 \) as monomial, binomial, trinomial, or zero polynomial.
- Find the degree of \( 2x^2y + 3xy^2 + 5 \).
- Factor \( x^2 + 7x + 10 \).
- State whether \( 5x^{-2} + 4 \) is a polynomial or not.
Classroom Revision Tip
A simple trick to quickly classify the degree of a polynomial: check only the highest exponent among all terms — that’s the degree! Vedantu’s teachers use this tip in live classes for quick board revision.
Further Learning and Practice
For more detailed concepts, solved examples, and advanced tricks, explore Polynomials - Full Topic and Factoring Polynomials. Practicing from Remainder Theorem and Factor Theorem sections helps boost confidence for board exams.
We explored polynomials for class 10 — from their definition, classification, degree, formulas, examples, and ties to other topics. Continue practicing with Vedantu to become confident in solving challenging questions using this concept!
FAQs on Polynomials for Class 10: Concepts, Types, and Formulas
1. What is a polynomial for class 10?
In Class 10 mathematics, a polynomial is an algebraic expression made up of variables, coefficients, and non-negative integer exponents. It is written in the form:
$a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$
where $a_n, a_{n-1}, ..., a_0$ are real numbers (coefficients), $x$ is a variable, and $n$ is a non-negative integer known as the degree of the polynomial. Polynomials are classified by their degree, such as linear (degree 1), quadratic (degree 2), and cubic (degree 3).
2. Is polynomial class 10 hard?
The difficulty of the polynomial chapter in Class 10 depends on a student's understanding of algebraic basics and practice with polynomial concepts. With clear explanations, regular practice, and support from teachers and platforms like Vedantu, many students find the chapter manageable. Vedantu offers structured courses, expert guidance, and targeted practice materials to help students excel in polynomials.
3. What is polynomial class 10 project?
A polynomial project for Class 10 typically involves exploring real-life applications, properties, and types of polynomials. Students might be required to:
- Research on polynomial uses in science, engineering, and everyday life
- Represent polynomials graphically and algebraically
- Demonstrate polynomial operations such as addition, subtraction, and multiplication
4. What is the formula to find the polynomial class 10?
In Class 10, the general formula for a polynomial in variable $x$ is:
$P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$
Here, each $a_i$ ($i = 0, 1, ..., n$) is a coefficient, $x$ is the variable, and $n$ is the degree of the polynomial. Specific formulae like the Remainder Theorem or Factor Theorem are also included in the Class 10 syllabus to solve related problems efficiently.
5. What are the important topics in polynomials for Class 10 exams?
Key polynomial topics for Class 10 board exams include:
- Types of polynomials (linear, quadratic, cubic)
- Zeroes of a polynomial
- Relationship between zeroes and coefficients
- Factorization of polynomials
- Remainder Theorem and Factor Theorem
- Graphical representation of polynomials
6. How can Vedantu help me understand polynomials better in Class 10?
Vedantu supports students in mastering polynomials for Class 10 through:
- Interactive live classes with subject experts
- Comprehensive study materials aligned with the latest curriculum
- Practice questions with step-by-step solutions
- Doubt-solving sessions and real-time guidance
- Revision notes and mock tests for exam preparation
7. What are the applications of polynomials in real life for Class 10 students?
Polynomials have practical applications in various fields relevant to Class 10, such as:
- Calculating areas and volumes in geometry
- Physics equations of motion (e.g., $s = ut + \frac{1}{2}at^2$)
- Economics (profit/loss predictions)
- Engineering (structural and velocity calculations)
- Computer science (algorithm development)
8. How do you find the zeroes of a quadratic polynomial in Class 10?
To find the zeroes of a quadratic polynomial ($ax^2 + bx + c$), students use these methods:
- Factorization: Express the quadratic as $(x - \alpha)(x - \beta)$ and solve for $x$.
- Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Completing the square
9. What is the relationship between zeroes and coefficients of a polynomial?
For a quadratic polynomial $ax^2 + bx + c$ with zeroes $\alpha$ and $\beta$:
- Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$
- Product of zeroes: $\alpha \times \beta = \frac{c}{a}$
