A quadratic polynomial is an algebraic expression of degree 2 represented as ax² + bx + c where a b and c are constants and a ≠ 0. It forms a parabola when graphed and its solutions are called roots. The roots can be found using factorization completing the square or the quadratic formula. It is widely used in real-life applications like physics engineering and economics.
Class 10 Maths Chapter 2 Polynomials focuses on understanding zeros and coefficients of polynomials along with division algorithms. It covers linear quadratic and cubic polynomials. This chapter prepares students for advanced algebraic concepts and problem-solving techniques essential for CBSE Exam 2024-25. Mastery of polynomials ensures a strong foundation in algebra and its applications in real-life scenarios and higher studies.
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Building a Quadratic Polynomial with Given Zeros
Step 1: Defining Zeros
– Given zeros: 4 and -3
– The polynomial shall thus be in the form: (x – 4)(x + 3)
Step 2: Algebraic Expansion
(x – 4)(x + 3) = x² + 3x – 4x – 12
= x² – x – 12
Step 3: Checking Zero Properties
Let’s test if the zeros hold:
– When x = 4:
4² – 4 – 12 = 16 – 4 – 12 = 0 OK
– When x = -3:
(-3)² – (-3) – 12 = 9 + 3 – 12 = 0 OK
Step 4: Coefficient Analysis
– Coefficient of x²: 1
– Coefficient of x: -1
– Constant term: -12
Mathematical Insights:
– The zeros of a quadratic define its form
– The general form illustrates how the zeros are connected to the coefficients of the polynomial
– Vieta’s formulas support the connection between coefficients and zeros
Conclusion:
The quadratic polynomial whose zeros are 4 and -3 is x² – x – 12.
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