If the sum and product of the zeros of a quadratic polynomial are 3 and -10, respectively, then the quadratic polynomial is:
The sum and product of the zeros of a quadratic polynomial are related to its coefficients. For a polynomial ax² + bx + c = 0 the sum of zeros is -b/a and the product of zeros is c/a. These relationships help in finding zeros or forming polynomials when zeros are given ensuring clarity in solving equations and understanding their properties.
Class 10 Maths Chapter 2 Polynomials covers key concepts like zeros of polynomials relationships between zeros and coefficients and division algorithm for polynomials. It emphasizes quadratic and cubic polynomials and their applications. This chapter prepares students for solving problems logically and scoring well in the CBSE Exam 2024-25 while building a strong foundation for advanced algebra and calculus essential for future studies.
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Step 1: Understanding Zeros and Quadratic Form
– A quadratic polynomial has the general form: x² + bx + c
– Its zeros are the roots that make the polynomial equal to zero
– Let the zeros be p and q
Step 2: Given Conditions
– Sum of zeros (p + q) = 3
– Product of zeros (p * q) = -10
Step 3: Relationship to Quadratic Coefficients
In the standard form x² + bx + c:
– b = -(sum of zeros)
– c = product of zeros
Step 4: Calculating Coefficients
– b = -(p + q) = -3
– c = p * q = -10
Step 5: Constructing the Quadratic Polynomial
The polynomial becomes:
x² – 3x – 10
Verification:
– Coefficient of x²: 1
– Coefficient of x: -3 (negative of zero sum)
– Constant term: -10 (product of zeros)
Mathematical Reasoning:
The coefficients directly reflect the given conditions about the zeros:
– Sum of zeros: p + q = 3
– Product of zeros: p * q = -10
Conclusion:
The quadratic polynomial is x² – 3x – 10.
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