CBSE Class 10 Mathematics (2026–27)
Chapter 2: Polynomials
20 Important Questions and Answers
Q1. What is a polynomial? Explain with examples.
Answer:
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where the powers of variables are non-negative integers. Examples include (x^2 + 3x + 2), (5x^3 – 2x + 7), and (4). Expressions such as (1/x) or (x^{-2}) are not polynomials because they contain negative powers of the variable. Polynomials are classified according to their degree. They are widely used in mathematics to represent relationships between quantities and solve equations. Understanding polynomials is important because they form the basis of higher algebra and many real-life applications in science and engineering.
Q2. Differentiate between monomial, binomial, and trinomial.
Answer:
Polynomials are classified according to the number of terms they contain. A monomial has only one term, such as (5x^2) or (7). A binomial has two unlike terms, such as (x+3) or (2x^2-5). A trinomial has three unlike terms, such as (x^2+2x+1). The terms are separated by plus or minus signs. These classifications help in understanding polynomial operations and factorization. For example, factorization methods often transform trinomials into products of binomials. Recognizing the type of polynomial makes solving algebraic problems easier and more systematic.
Q3. What is the degree of a polynomial? Explain with examples.
Answer:
The degree of a polynomial is the highest power of the variable present in the polynomial. It determines the type and behavior of the polynomial. For example, in (3x^4+2x^2+1), the highest power is 4, so the degree is 4. In (5x^3-2x+7), the degree is 3. A constant polynomial like 9 has degree 0. The degree helps classify polynomials into linear, quadratic, cubic, and higher-degree polynomials. It also indicates the maximum number of zeros a polynomial can have. Understanding the degree is essential for solving polynomial equations and graph-related problems.
Q4. Define a linear polynomial and give examples.
Answer:
A linear polynomial is a polynomial whose highest power of the variable is 1. Its general form is (ax+b), where (a\neq0). Examples include (2x+3), (5x-7), and (x+1). Linear polynomials represent straight-line relationships when plotted on a graph. They have exactly one zero, which can be found by setting the polynomial equal to zero. For example, the zero of (x+2) is (-2). Linear polynomials are the simplest form of polynomials and are frequently used in algebra, coordinate geometry, and practical situations involving proportional relationships.
Q5. What is a quadratic polynomial? Give examples.
Answer:
A quadratic polynomial is a polynomial whose highest power of the variable is 2. Its general form is (ax^2+bx+c), where (a\neq0). Examples include (x^2+3x+2), (2x^2-5x+1), and (4x^2+7). A quadratic polynomial can have at most two zeros. The graph of a quadratic polynomial is a parabola. These polynomials are important in solving problems involving area, motion, and optimization. Factorization and the quadratic formula are commonly used methods to find their zeros. Quadratic polynomials form a major part of algebra and higher mathematics.
Q6. What is a cubic polynomial? Explain with examples.
Answer:
A cubic polynomial is a polynomial whose highest power of the variable is 3. Its general form is (ax^3+bx^2+cx+d), where (a\neq0). Examples are (x^3-2x+1) and (2x^3+3x^2-5). A cubic polynomial can have at most three zeros. The graph of a cubic polynomial usually changes direction and may intersect the x-axis up to three times. Cubic polynomials are useful in advanced algebra and real-life applications such as engineering and physics. Understanding cubic polynomials helps students prepare for higher-level mathematical concepts and problem-solving techniques.
Q7. What are the zeros of a polynomial?
Answer:
The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. To find a zero, substitute a value into the polynomial and check whether the result is zero. For example, for (p(x)=x-3), substituting (x=3) gives (0), so 3 is a zero of the polynomial. Zeros are important because they represent the points where the graph of the polynomial intersects the x-axis. The number of zeros of a polynomial can never exceed its degree. Finding zeros helps in solving equations and understanding polynomial behavior.
Q8. Find the zero of the polynomial (x+5).
Answer:
To find the zero of the polynomial (p(x)=x+5), we set the polynomial equal to zero:
[
x+5=0
]
Subtracting 5 from both sides gives:
[
x=-5
]
Therefore, (-5) is the zero of the polynomial. We can verify this by substituting (x=-5):
[
(-5)+5=0
]
Since the result is zero, the value is correct. A linear polynomial always has one zero. The zero represents the point where the graph of the polynomial intersects the x-axis. Understanding how to find zeros is a basic and important skill in algebra.
Q9. Verify whether 2 is a zero of the polynomial (x^2-4).
Answer:
To verify whether 2 is a zero of the polynomial (p(x)=x^2-4), substitute (x=2):
[
p(2)=2^2-4
]
[
=4-4
]
[
=0
]
Since the value of the polynomial becomes zero, 2 is a zero of the polynomial. Similarly, (-2) is also a zero because:
[
(-2)^2-4=4-4=0
]
Thus, the polynomial has two zeros, namely 2 and -2. Verification of zeros is useful in factorization and solving polynomial equations. It helps students understand the relationship between algebraic expressions and their roots.
Q10. Explain the geometrical meaning of the zero of a polynomial.
Answer:
The geometrical meaning of the zero of a polynomial is the point where its graph intersects the x-axis. At these points, the value of the polynomial is zero. For a linear polynomial, the graph is a straight line and it intersects the x-axis at one point. For a quadratic polynomial, the graph is a parabola and may intersect the x-axis at two, one, or no points. These intersection points represent the zeros of the polynomial. The graphical interpretation helps students visualize solutions of polynomial equations and understand the connection between algebra and geometry.
Q11. State the relationship between zeros and coefficients of a quadratic polynomial.
Answer:
For a quadratic polynomial (ax^2+bx+c), let the zeros be (\alpha) and (\beta). Then:
[
\alpha+\beta=-\frac{b}{a}
]
and
[
\alpha\beta=\frac{c}{a}
]
This relationship connects the zeros directly with the coefficients of the polynomial. It is useful when the zeros are known and the polynomial needs to be formed, or when the coefficients are known and properties of the zeros are required. These formulas simplify many algebraic calculations and are frequently used in examinations. They also provide a deeper understanding of quadratic equations and factorization.
Q12. Find the sum and product of zeros of (x^2-5x+6).
Answer:
Given the quadratic polynomial:
[
x^2-5x+6
]
Here, (a=1), (b=-5), and (c=6).
Using the formulas:
[
\text{Sum of zeros}=-\frac{b}{a}
]
[
=-\frac{-5}{1}=5
]
[
\text{Product of zeros}=\frac{c}{a}
]
[
=\frac{6}{1}=6
]
Therefore, the sum of the zeros is 5 and the product of the zeros is 6. These values can also be verified using the actual zeros, which are 2 and 3. Their sum is 5 and their product is 6, confirming the relationship.
Q13. Form a quadratic polynomial whose zeros are 2 and 5.
Answer:
Let the zeros be 2 and 5. A quadratic polynomial with zeros (\alpha) and (\beta) is:
[
x^2-(\alpha+\beta)x+\alpha\beta
]
Substituting the given zeros:
[
x^2-(2+5)x+(2\times5)
]
[
=x^2-7x+10
]
Therefore, the required quadratic polynomial is:
[
x^2-7x+10
]
We can verify this by factorization:
[
x^2-7x+10=(x-2)(x-5)
]
Thus, 2 and 5 are indeed the zeros of the polynomial. Forming polynomials from given zeros is an important application of the relationship between zeros and coefficients.
Q14. Explain the Remainder Theorem.
Answer:
The Remainder Theorem states that when a polynomial (p(x)) is divided by ((x-a)), the remainder is equal to (p(a)). This theorem provides a quick way to find the remainder without performing long division. For example, if (p(x)=x^2+3x+2) and it is divided by ((x-1)), then:
[
p(1)=1+3+2=6
]
Hence, the remainder is 6. The theorem is useful in checking divisibility and finding zeros of polynomials. It simplifies calculations and forms the basis for the Factor Theorem, which is frequently used in polynomial factorization.
Q15. State and explain the Factor Theorem.
Answer:
The Factor Theorem states that ((x-a)) is a factor of a polynomial (p(x)) if and only if (p(a)=0). In other words, when substituting (a) into the polynomial gives zero, ((x-a)) divides the polynomial exactly without leaving any remainder. For example, for (p(x)=x^2-4):
[
p(2)=4-4=0
]
Therefore, ((x-2)) is a factor of the polynomial. The Factor Theorem helps in finding factors and zeros efficiently. It is widely used in algebra for simplifying expressions and solving polynomial equations.
Q16. Check whether ((x-1)) is a factor of (x^2-1).
Answer:
Using the Factor Theorem, let:
[
p(x)=x^2-1
]
To check whether ((x-1)) is a factor, substitute (x=1):
[
p(1)=1^2-1
]
[
=1-1
]
[
=0
]
Since the value obtained is zero, ((x-1)) is a factor of the polynomial. In fact:
[
x^2-1=(x-1)(x+1)
]
This confirms the result. The Factor Theorem provides a quick and effective method for determining whether a given expression is a factor of a polynomial.
Q17. Why can a quadratic polynomial have at most two zeros?
Answer:
A quadratic polynomial has degree 2, meaning the highest power of the variable is 2. According to the fundamental property of polynomials, a polynomial can have at most as many zeros as its degree. Therefore, a quadratic polynomial can have at most two zeros. Graphically, the parabola representing a quadratic polynomial can intersect the x-axis at two points, one point, or not intersect it at all. These intersections correspond to its zeros. This concept helps students understand the relationship between the degree of a polynomial and the number of possible solutions to the equation.
Q18. Explain constant and zero polynomials.
Answer:
A constant polynomial is a polynomial whose value remains fixed and does not contain any variable. Examples are 5, -3, and 10. A non-zero constant polynomial has degree 0. A zero polynomial is a polynomial in which all coefficients are zero, represented by 0. Unlike other polynomials, the degree of the zero polynomial is not defined. Constant and zero polynomials are special cases of polynomials and are important in algebraic operations. Understanding these concepts helps students classify polynomials correctly and avoid common mistakes while determining degrees and performing calculations.
Q19. How are polynomials used in real life?
Answer:
Polynomials are widely used in everyday life and various fields of science and technology. Engineers use polynomials to design structures and machines. Physicists use them to represent motion and physical laws. Economists use polynomial equations to model profit, cost, and demand relationships. In computer graphics, polynomials help create curves and animations. They are also used in statistics, architecture, and data analysis. Because polynomials can represent complex relationships mathematically, they are powerful tools for prediction and problem-solving. Learning polynomials in school provides a foundation for many advanced applications in higher education and professional careers.
Q20. What are the main classifications of polynomials based on degree?
Answer:
Polynomials are classified according to their degree, which is the highest power of the variable. A polynomial of degree 0 is called a constant polynomial. Degree 1 polynomials are linear polynomials, such as (x+2). Degree 2 polynomials are quadratic polynomials, such as (x^2+3x+1). Degree 3 polynomials are cubic polynomials, such as (x^3-2x+4). Polynomials of degree 4 and above are called higher-degree polynomials. This classification helps students identify polynomial types and determine their properties, such as the maximum number of zeros and the methods used for solving related equations.