Polynomials Class 10 – Zeros, Factorization & Remainder Theorem (Complete Guide)
Polynomials is one of the most important chapters in CBSE Class 10 Mathematics. It forms the foundation for quadratic equations, higher algebra, and even calculus in higher classes. Understanding zeros of polynomials, factorization techniques, and the Remainder Theorem is essential for scoring full marks in board exams.
In this detailed guide, we will cover:
- Definition and types of polynomials
- Zeros of polynomials
- Relationship between zeros and coefficients
- Factorization methods
- Remainder Theorem
- Multiple solved problems with step-by-step explanations
1. What is a Polynomial?
A polynomial in one variable x is an algebraic expression of the form:
a₀ + a₁x + a₂x² + a₃x³ + … + aₙxⁿ
where n is a non-negative integer and a₀, a₁, a₂… are real numbers.
Types of Polynomials Based on Degree
- Constant Polynomial: Degree 0 (Example: 5)
- Linear Polynomial: Degree 1 (Example: 2x + 3)
- Quadratic Polynomial: Degree 2 (Example: x² + 5x + 6)
- Cubic Polynomial: Degree 3 (Example: x³ − 2x² + x)
2. Zeros of a Polynomial
A zero of a polynomial is a value of x for which the polynomial becomes zero. If p(x) = 0, then x is called a zero of the polynomial.
Example 1: Find the zero of p(x) = 2x + 4
2x + 4 = 0
2x = -4
x = -2
Zero = -2
Graphical Meaning of Zeros
The zeros of a polynomial are the points where its graph cuts or touches the x-axis.
3. Relationship Between Zeros and Coefficients
For quadratic polynomial ax² + bx + c:
- Sum of zeros = -b/a
- Product of zeros = c/a
Example 2: Find sum and product of zeros of 3x² − 5x − 2
a = 3, b = -5, c = -2
Sum = -(-5)/3 = 5/3
Product = -2/3
4. Finding Polynomial When Zeros Are Given
If zeros are α and β, then quadratic polynomial is:
x² − (α + β)x + (αβ)
Example 3: Form polynomial whose zeros are 2 and 5
Sum = 7, Product = 10
Polynomial = x² − 7x + 10
5. Factorization of Polynomials
Factorization means expressing a polynomial as a product of simpler polynomials.
Method 1: Splitting the Middle Term
Example 4: Factorize x² + 7x + 10
Find two numbers whose sum is 7 and product is 10 → 5 and 2
x² + 5x + 2x + 10
x(x + 5) + 2(x + 5)
(x + 5)(x + 2)
Method 2: Using Identities
Example 5: Factorize x² − 9
= x² − 3²
= (x − 3)(x + 3)
6. Remainder Theorem
If a polynomial p(x) is divided by (x − a), then the remainder is p(a).
Example 6: Find remainder when p(x) = x³ − 2x² + 4 is divided by (x − 2)
According to Remainder Theorem:
Remainder = p(2)
= 8 − 8 + 4
= 4
Remainder = 4
Factor Theorem
If p(a) = 0, then (x − a) is a factor of p(x).
Example 7: Check whether (x − 1) is factor of x³ − 3x² + 2x
p(1) = 1 − 3 + 2 = 0
Since p(1) = 0, (x − 1) is a factor.
7. Division Algorithm for Polynomials
Dividend = Divisor × Quotient + Remainder
Degree of remainder must be less than divisor.
8. Important Board Exam Questions (Practice Set)
Question 1: Find quadratic polynomial whose sum and product are -3 and 2.
Polynomial = x² + 3x + 2
Question 2: Find remainder when 2x³ − 5x + 1 is divided by (x + 1).
Remainder = p(-1)
= -2 + 5 + 1 = 4
Question 3: Factorize 2x² − 7x + 3
= 2x² − 6x − x + 3
= 2x(x − 3) −1(x − 3)
= (2x − 1)(x − 3)
Common Mistakes Students Make
- Forgetting negative sign in sum of zeros formula
- Incorrect splitting of middle term
- Not substituting correctly in Remainder Theorem
- Confusing Factor Theorem and Remainder Theorem
Exam Tips for Full Marks
✔ Verify factorization by multiplying back
✔ Practice 10 Remainder Theorem problems
✔ Revise sum-product formulas daily
Conclusion
Polynomials is a high-scoring and concept-based chapter in Class 10 Algebra. With strong understanding of zeros, factorization, and Remainder Theorem, students can easily solve complex algebraic problems. Regular practice and formula revision are the keys to mastering this topic.