The number of polynomials having given zeros depends on the degree and coefficients. For two specific zeros like -3 and 5 infinite polynomials can exist as the coefficient k in P(x) = k(x + 3)(x – 5) can take any non-zero value. This flexibility ensures there are more than 3 or infinite polynomials possible.
Class 10 Maths Chapter 2 Polynomials covers key concepts like zeros of polynomials relationships between zeros and coefficients and division algorithm. It focuses on linear quadratic and cubic polynomials. This chapter strengthens algebraic skills essential for CBSE Exam 2024-25 and helps in solving real-life problems. Understanding these topics builds a strong foundation for advanced mathematics and ensures better performance in exams and competitive tests.
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Building Polynomials with Specified Zeros
Step 1: Learning Polynomial Building
– Provided zeros: -3 and 5
– Simple polynomial form: (x + 3)(x – 5)
– Expanding: x² – 2x – 15
Step 2: Freedom Degree
Polynomials may be formed by multiplying the simple form by any non-zero constant.
Possible Polynomials:
1. x² – 2x – 15
2. 2x² – 4x – 30
3. 3x² – 6x – 45
General Form:
For any non-zero constant k:
k(x² – 2x – 15)
Mathematical Insight:
– The k is the parameter for infinite scaling of the polynomial
– Each scale produces a different polynomial with the same zeros
– These are actually the same polynomial to a constant factor
Step 3: Counting Polynomials
– We can make polynomials with these zeros by multiplying the fundamental form
by any real number
– This produces an infinite number of polynomials
Conclusion
There are MORE THAN 3 polynomials with zeros -3 and 5.
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