Introduction: Polynomials (Class X)
Polynomials is the second chapter of Class X Mathematics and plays an important role in developing algebraic understanding. In this chapter, students learn about algebraic expressions involving variables and constants, combined using addition, subtraction, and multiplication.
The chapter focuses on different types of polynomials based on their degree, such as linear, quadratic, and cubic polynomials. Students also study the concept of zeros of a polynomial and understand their graphical representation as points where the graph intersects the x-axis.
An important part of this chapter is the relationship between the zeros and coefficients of a quadratic polynomial. The Division Algorithm for Polynomials is also introduced, which helps in dividing one polynomial by another.
A strong understanding of polynomials is essential for later chapters like Quadratic Equations, Linear Equations, and Coordinate Geometry. This chapter is highly important for board examinations and competitive exams.
Polynomials – 30 MCQ Quiz (Class X)
1. A polynomial of degree 1 is called:
2. The degree of polynomial 3x² + 5x + 7 is:
3. How many zeros can a quadratic polynomial have at most?
4. Zeros of a polynomial are the points where its graph cuts:
5. The degree of a constant non-zero polynomial is:
6. A polynomial having three zeros is of degree:
7. Which of the following is a quadratic polynomial?
8. The sum of zeros of ax² + bx + c is:
9. The product of zeros of ax² + bx + c is:
10. Number of zeros of a linear polynomial is:
11. Which polynomial has no zero?
12. The degree of zero polynomial is:
13. Graph of a linear polynomial is a:
14. A cubic polynomial can have at most:
15. The polynomial division algorithm is similar to:
16. If p(x) = x² − 1, then p(1) equals:
17. The zeros of x² − 4 are:
18. Which polynomial has degree 3?
19. The graph of a quadratic polynomial is a:
20. Which of the following is NOT a polynomial?
21. A polynomial with degree zero is:
22. The number of zeros of x² + 1 is:
23. If sum of zeros is 3 and product is 2, polynomial is:
24. Degree of polynomial 0 is:
25. Which is a linear polynomial?
26. The product of zeros of x² − 5x + 6 is:
27. A polynomial of degree n has at most:
28. Which polynomial has exactly one zero?
29. If p(0) = 3, constant term is:
30. If α and β are the zeros of the polynomial x² − 7x + 10, then α + β is:
Polynomials – 50 Important Facts (Class X)
- A polynomial is an algebraic expression made up of variables and constants.
- Only non-negative integer powers of variables are allowed in polynomials.
- x² + 3x − 5 is an example of a polynomial.
- 1/x is not a polynomial.
- The highest power of the variable is called the degree of a polynomial.
- A constant non-zero polynomial has degree 0.
- A linear polynomial has degree 1.
- A quadratic polynomial has degree 2.
- A cubic polynomial has degree 3.
- The degree of the zero polynomial is not defined.
- A linear polynomial has exactly one zero.
- A quadratic polynomial has at most two zeros.
- A cubic polynomial has at most three zeros.
- The zero of a polynomial is the value of x for which p(x) = 0.
- Zeros of a polynomial are also called roots.
- Graphically, zeros are the points where the graph cuts the x-axis.
- The graph of a linear polynomial is a straight line.
- The graph of a quadratic polynomial is a parabola.
- The graph of a cubic polynomial is a curve.
- A constant polynomial (non-zero) has no zero.
- The polynomial ax² + bx + c is a quadratic polynomial.
- The sum of zeros of ax² + bx + c is −b/a.
- The product of zeros of ax² + bx + c is c/a.
- These relationships are very important for board exams.
- A polynomial can be constructed if sum and product of zeros are given.
- p(0) gives the constant term of a polynomial.
- If p(a) = 0, then a is a zero of the polynomial.
- Factor theorem helps in finding factors of polynomials.
- If (x − a) is a factor of p(x), then p(a) = 0.
- Polynomial division is similar to long division of numbers.
- The division algorithm for polynomials is: p(x) = g(x)q(x) + r(x).
- The degree of remainder is always less than the degree of divisor.
- Division algorithm helps verify division results.
- A polynomial may have equal zeros.
- x² − 4 has two zeros: 2 and −2.
- x² + 1 has no real zero.
- The coefficient of the highest power term is called the leading coefficient.
- Every polynomial can be represented graphically.
- Polynomials form the base for Quadratic Equations.
- This chapter is very important for board and competitive exams.
