Class 10 Maths MCQ Test – Polynomial (Chapter 2) | Full Chapter Test
Class 10 Mathematics – Chapter 2: Polynomials (Detailed Notes)
These notes are designed for CBSE, GSEB and board exam preparation. It includes definitions, formulas, properties, and exam-oriented concepts.
1. What is a Polynomial?
A polynomial is an algebraic expression consisting of variables and coefficients, where the powers of variables are whole numbers (0, 1, 2, 3, ...).
General form of a polynomial in one variable:
\( p(x) = a_n x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)
Where:
- \( a_n, a_{n-1}, ..., a_0 \) are real numbers
- \( a_n \neq 0 \)
- \( n \) is a non-negative integer
2. Types of Polynomials
- Constant Polynomial: Degree = 0 (e.g. 5)
- Linear Polynomial: Degree = 1 (e.g. \( ax + b \))
- Quadratic Polynomial: Degree = 2 (e.g. \( ax^2 + bx + c \))
- Cubic Polynomial: Degree = 3 (e.g. \( ax^3 + bx^2 + cx + d \))
3. Degree of a Polynomial
The highest power of the variable in a polynomial is called its degree.
Example:
- \( 3x^2 + 2x + 1 \) → Degree = 2
- \( 5x^3 + x \) → Degree = 3
4. Zeroes of a Polynomial
A value of \( x \) for which \( p(x) = 0 \) is called a zero or root of the polynomial.
Graphically, zeroes are the points where the graph of polynomial intersects the x-axis.
Example: If \( p(x) = x - 3 \), then zero is \( x = 3 \).
5. Remainder Theorem
If a polynomial \( p(x) \) is divided by \( (x - a) \), then the remainder is \( p(a) \).
\( \text{Remainder} = p(a) \)
Example: If \( p(x) = x^2 + 2x + 1 \), then remainder when divided by \( (x - 1) \) is:
\( p(1) = 1 + 2 + 1 = 4 \)
6. Factor Theorem
A polynomial \( (x - a) \) is a factor of \( p(x) \) if and only if \( p(a) = 0 \).
If \( p(a) = 0 \Rightarrow (x - a) \) is a factor
7. Division Algorithm for Polynomials
For polynomials \( p(x) \) and \( g(x) \), where \( g(x) \neq 0 \):
\( p(x) = g(x) \cdot q(x) + r(x) \)
Where:
- \( p(x) \) = Dividend
- \( g(x) \) = Divisor
- \( q(x) \) = Quotient
- \( r(x) \) = Remainder
Condition: Degree of \( r(x) \) < Degree of \( g(x) \)
8. Zeroes of Quadratic Polynomial
For a quadratic polynomial \( ax^2 + bx + c \), zeroes are roots of equation:
\( ax^2 + bx + c = 0 \)
If \( \alpha \) and \( \beta \) are zeroes, then:
- \( \alpha + \beta = \frac{-b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)
9. Relation Between Zeroes and Coefficients
For quadratic polynomial \( ax^2 + bx + c \):
Sum of zeroes:
\( \alpha + \beta = -\frac{b}{a} \)
Product of zeroes:
\( \alpha \beta = \frac{c}{a} \)
10. Factorisation of Polynomials
Methods of factorisation:
- Taking common factor
- Splitting middle term
- Using identities
Important Identities:
- \( (a+b)^2 = a^2 + 2ab + b^2 \)
- \( (a-b)^2 = a^2 - 2ab + b^2 \)
- \( a^2 - b^2 = (a-b)(a+b) \)
11. Graph of a Polynomial
The graph of a polynomial helps in identifying zeroes:
- Linear → Straight line
- Quadratic → Parabola
- Cubic → S-shaped curve
Zeroes are points where graph cuts x-axis.
⬇️ Practice MCQs and numericals after revising these notes
Class 10 Maths
MCQ Test – Polynomials (Full Chapter Test)
Class 10 Maths Chapter 2: Polynomials
Polynomials is a fundamental chapter in Class 10 Mathematics. It forms the base for higher algebra and plays an important role in board exams. Understanding this chapter helps students analyze algebraic expressions, graphs, and equations effectively.
Learning Outcomes
After completing this chapter, students will be able to:
- Understand the concept of polynomials and their degrees.
- Identify types of polynomials such as linear, quadratic, and cubic.
- Find zeros of a polynomial graphically and algebraically.
- Understand the relationship between zeros and coefficients.
- Apply the Factor Theorem and Remainder Theorem.
- Factorize polynomials using different methods.
- Analyze graphs of polynomials and understand their behavior.
Important Concepts
- Polynomial: An expression consisting of variables and coefficients.
- Degree: The highest power of the variable.
- Zeros: Values of x where polynomial becomes zero.
- Graph: The number of zeros equals the number of points where graph cuts x-axis.
- Factor Theorem: If f(a) = 0, then (x - a) is a factor of polynomial.
Real Life Applications
Polynomials are widely used in real life and various fields:
- Engineering: Used in designing structures and curves.
- Economics: Used to model profit and cost functions.
- Physics: Helps in motion equations and trajectory calculations.
- Computer Graphics: Used to design curves and animations.
- Business: Used to predict trends and growth patterns.
Common Mistakes Students Make
- Confusing degree of polynomial with number of terms.
- Incorrectly identifying zeros from graph.
- Forgetting sign changes while factorization.
- Applying Factor Theorem incorrectly.
- Skipping steps in long division method.
- Not verifying answers after solving.
Conclusion
The chapter “Polynomials” is highly scoring and concept-based. With regular practice of graphs, factorization, and theorem-based questions, students can easily master this chapter. Focus on clarity of concepts and avoid common mistakes to score high marks in exams.
Class 10 Maths Chapter 2: Polynomials
10 Detailed Solved Examples
Find the zeros of the polynomial f(x) = x² − 5x + 6.
Solution:
Factorize: x² − 5x + 6 = (x − 2)(x − 3)
So, zeros are 2 and 3.
Find sum and product of zeros of 2x² − 7x + 3.
Solution:
Sum = −b/a = 7/2
Product = c/a = 3/2
Verify relationship between zeros and coefficients for x² − 3x + 2.
Solution:
Zeros = 1, 2
Sum = 1 + 2 = 3 = −(−3)/1 ✔
Product = 1 × 2 = 2 ✔
Check whether (x − 2) is a factor of f(x) = x³ − 8.
Solution:
f(2) = 8 − 8 = 0
Hence, (x − 2) is a factor.
Find remainder when x³ − 3x² + 4 is divided by (x − 1).
Solution:
f(1) = 1 − 3 + 4 = 2
Remainder = 2
How many zeros does a quadratic polynomial have?
Solution:
A quadratic polynomial can have maximum 2 zeros.
Factorize x³ − 3x² − 4x + 12.
Solution:
Group terms:
x²(x − 3) − 4(x − 3)
= (x − 3)(x² − 4)
= (x − 3)(x − 2)(x + 2)
Verify identity: (x + y)² = x² + 2xy + y² for x = 2, y = 3.
Solution:
LHS = (2 + 3)² = 25
RHS = 4 + 12 + 9 = 25 ✔
Find one zero of x³ − x² − x + 1.
Solution:
Try x = 1:
f(1) = 1 − 1 − 1 + 1 = 0
So, 1 is a zero.
If a graph cuts x-axis at 3 points, what is the degree?
Solution:
Maximum zeros = degree
So degree ≥ 3.
10 Detailed FAQs
A polynomial is an algebraic expression consisting of variables, coefficients, and powers of variables.
The highest power of the variable in the polynomial.
Values of x that make the polynomial equal to zero.
If f(a) = 0, then (x − a) is a factor of polynomial.
Remainder when dividing by (x − a) is f(a).
Number of x-axis intersections equals number of zeros.
No, maximum zeros = degree of polynomial.
A polynomial of degree 2.
It forms base for algebra, calculus, and higher mathematics.
Practice factorization, theorems, and graph-based questions regularly.
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