Definition Types Standard Form and Solved Examples of Polynomials for Class 10
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 Polynomial for Class 10 Complete Guide with Concepts and Practice
1. What is a polynomial in Class 10 Maths?
A polynomial is an algebraic expression made up of variables and coefficients combined using addition, subtraction, and multiplication with non-negative integer powers of variables.
- General form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
- Here, aₙ, aₙ₋₁, ..., a₀ are constants and n is a non-negative integer.
- Example: 3x² + 5x − 7 is a polynomial.
2. What is the degree of a polynomial?
The degree of a polynomial is the highest power of the variable in the expression.
- In 4x³ + 2x² − x + 6, the highest power is 3.
- So, the degree is 3.
3. How do you find the zeroes of a polynomial?
The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero.
- Step 1: Set the polynomial equal to 0.
- Step 2: Solve the equation.
4. What is the relationship between zeroes and coefficients of a quadratic polynomial?
For a quadratic polynomial ax² + bx + c, the relationship between zeroes and coefficients is given by standard formulas.
- Sum of zeroes = −b/a
- Product of zeroes = c/a
5. How do you divide one polynomial by another?
You can divide one polynomial by another using the long division method or synthetic division.
- Arrange terms in descending powers.
- Divide the first term of the dividend by the first term of the divisor.
- Multiply and subtract.
- Repeat until the remainder degree is less than the divisor.
6. What is the Remainder Theorem in polynomials?
The Remainder Theorem states that when a polynomial p(x) is divided by (x − a), the remainder is p(a).
- Example: If p(x) = x² − 3x + 2 and divided by (x − 1),
- Find p(1) = 1 − 3 + 2 = 0.
7. What is the Factor Theorem in Class 10 Maths?
The Factor Theorem states that (x − a) is a factor of polynomial p(x) if and only if p(a) = 0.
- Substitute x = a in the polynomial.
- If the result is 0, then (x − a) is a factor.
8. How many zeroes can a polynomial have?
A polynomial of degree n can have at most n zeroes.
- A linear polynomial (degree 1) has at most 1 zero.
- A quadratic polynomial (degree 2) has at most 2 zeroes.
- A cubic polynomial (degree 3) has at most 3 zeroes.
9. What is a quadratic polynomial?
A quadratic polynomial is a polynomial of degree 2, generally written as ax² + bx + c where a ≠ 0.
- Its graph is a parabola.
- It can have 0, 1, or 2 real zeroes.
10. How do you form a quadratic polynomial if the zeroes are given?
If the zeroes of a quadratic polynomial are α and β, then the polynomial is formed as x² − (α + β)x + αβ.
- Step 1: Find the sum (α + β).
- Step 2: Find the product (αβ).
- Step 3: Substitute in the formula.
