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 in Class 10 Maths?
A polynomial in Class 10 Maths is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with only non-negative integer exponents. For example, 3x² + 2x - 5 is a polynomial.
2. How many types of polynomials are there in Class 10?
Polynomials are classified based on the number of terms and their degree. Based on the number of terms, we have:
- Monomials (one term)
- Binomials (two terms)
- Trinomials (three terms)
- Linear (degree 1)
- Quadratic (degree 2)
- Cubic (degree 3)
3. What is the degree of a polynomial?
The degree of a polynomial is the highest exponent of the variable in the polynomial. For example, in 5x³ + 2x² - 7x + 4, the degree is 3.
4. How do I factorize a quadratic polynomial?
Factorization of a quadratic polynomial involves expressing it as a product of two linear factors. Methods include:
- Splitting the middle term: Find two numbers whose sum is the coefficient of x and whose product is the product of the coefficient of x² and the constant term.
- Using the quadratic formula: Solve the equation ax² + bx + c = 0 to find the roots, then the factors are (x - root1)(x - root2).
5. Are polynomials important for board exams?
Yes, polynomials are a crucial topic in Class 10 board exams. They form the base for many higher-level algebraic concepts and appear frequently in both objective and subjective questions.
6. What is a zero polynomial?
A zero polynomial is a polynomial where all coefficients are zero. It is denoted as 0. It doesn't have a degree.
7. What are the zeroes of a polynomial?
The zeroes (or roots) of a polynomial are the values of the variable that make the polynomial equal to zero. Graphically, they represent the x-intercepts of the polynomial's graph.
8. How do I find the zeroes of a quadratic polynomial?
To find the zeroes of a quadratic polynomial, set the polynomial equal to zero and solve the resulting quadratic equation using methods like factoring, completing the square, or the quadratic formula.
9. What is the difference between a polynomial and an algebraic expression?
All polynomials are algebraic expressions, but not all algebraic expressions are polynomials. A polynomial has only non-negative integer exponents on its variables, whereas algebraic expressions can have variables raised to fractional or negative powers, or involve roots of variables.
10. How are polynomials used in real-life situations?
Polynomials have many real-world applications, including modeling curves in engineering, representing growth or decay in biological contexts, and describing projectile motion in physics.
11. What is the Remainder Theorem?
The Remainder Theorem states that when a polynomial P(x) is divided by (x - a), the remainder is P(a).
