Aim

To understand the relationship between the coefficients of a quadratic polynomial (ax² + bx + c) and the shape/position of its graph (a parabola), and to visually interpret its zeroes.

Objectives

• To plot graphs of various quadratic polynomials
• To analyze the effect of coefficients on the parabola's shape and position
• To identify zeroes of polynomials from their graphs
• To verify the relationship between zeroes and coefficients

Materials

• Graph paper
• Pencil and eraser
• Scale/Ruler
• Different colored pens

Tools

• Mathematical compass (optional)
• Calculator

Procedure

1. Selected various quadratic polynomials with different coefficient values
2. Created value tables for each polynomial by substituting x values
3. Plotted points on graph paper for each polynomial using appropriate scale
4. Joined the points with smooth curves to form parabolas
5. Analyzed the direction, width, and position of each parabola
6. Identified x-intercepts (zeroes) from the graphs
7. Verified the zeroes using algebraic methods
8. Compared the sum and product of zeroes with coefficient relationships

Observations

• When a > 0, the parabola opens upwards; when a < 0, it opens downwards
• Larger |a| values produce narrower parabolas
• The constant term c determines the y-intercept of the graph
• The coefficient b affects the position of the vertex along the x-axis
• The number of x-intercepts (zeroes) can be 0, 1, or 2
• For y = x² + 2x - 3, the zeroes are x = -3 and x = 1
• Sum of zeroes = -2 = -b/a, Product of zeroes = -3 = c/a

Conclusion

The graph of a quadratic polynomial is always a parabola. The coefficient a determines the direction and width of the parabola, b affects the position of the vertex, and c gives the y-intercept. The zeroes of the polynomial are the x-coordinates where the graph intersects the x-axis. The relationships α+β = -b/a and αβ = c/a were verified through this project.

Experience

This project helped me visualize the abstract concept of polynomials. Drawing the graphs made it easier to understand how changing coefficients affects the shape and position of the parabola. I learned to connect algebraic expressions with their geometric representations. The hands-on approach made the mathematical relationships more memorable and intuitive.

Aim

To verify the relationship between the zeroes and coefficients of a cubic polynomial and to use the Division Algorithm to find all zeroes when one is known.

Objectives

• To verify relationships between zeroes and coefficients of cubic polynomials
• To apply the Division Algorithm to factorize polynomials
• To find all zeroes of a cubic polynomial when one zero is known
• To connect algebraic and geometric representations of polynomials

Materials

• Pen and paper
• Calculator
• Chart paper for presentation

Tools

• Mathematical tables (if needed)

Procedure

1. Selected cubic polynomials that can be factorized easily
2. Found zeroes of each polynomial by factorization method
3. Recorded the zeroes and coefficients for each polynomial
4. Verified the relationships:
   • α + β + γ = -b/a
   • αβ + βγ + γα = c/a
   • αβγ = -d/a
5. Selected a cubic polynomial with one known zero
6. Applied the Division Algorithm to divide the polynomial by (x - known zero)
7. Factorized the quotient to find the remaining zeroes
8. Verified all zeroes by substituting in the original polynomial

Observations

• Cubic polynomials can have 1 or 3 real zeroes
• The relationships between zeroes and coefficients hold for all cubic polynomials
• The Division Algorithm is an efficient method to find all zeroes when one is known
• The graph of a cubic polynomial can intersect the x-axis at 1 or 3 points
• For x³ - 4x, the zeroes are -2, 0, and 2
• Sum of zeroes = 0 = -b/a, Sum of products = -4 = c/a, Product = 0 = -d/a

Conclusion

Cubic polynomials follow specific relationships between their zeroes and coefficients. The Division Algorithm provides a systematic method to find all zeroes when one zero is known. These relationships and methods connect the algebraic and geometric aspects of polynomials, enhancing our understanding of their behavior and properties.

Experience

Working with cubic polynomials helped me understand higher-degree algebraic expressions. The Division Algorithm was particularly interesting as it provided a structured approach to polynomial factorization. Seeing the consistent relationships between zeroes and coefficients across different examples reinforced my understanding of polynomial theory. This project strengthened my algebraic manipulation skills and deepened my appreciation for the interconnectedness of mathematical concepts.

Acknowledgement

I would like to express my sincere gratitude to my mathematics teacher, Mrs. Priya Mehta, for her guidance and support throughout this project. Her explanations and insights helped me understand the concepts clearly. I also thank my school for providing the necessary resources and my parents for their encouragement. Finally, I acknowledge the textbook authors whose work formed the foundation of this project.

References

• Mathematics Textbook for Class X - NCERT
• Comprehensive Mathematics - R.D. Sharma
• Mathematics Class X - R.S. Aggarwal
• Khan Academy - Online resources on polynomials
• Byju's Learning App - Interactive polynomial lessons