Class 10 Exam > Class 10 Questions > A polynomial of degree n has:a)Only one zerob...
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A polynomial of degree n has:
- a)Only one zero
- b)At least n zeroes
- c)More than n zeroes
- d)At most n zeroes
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More ...
Maximum number of zeroes of a polynomial = Degree of the polynomial
Most Upvoted Answer
A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More ...
Understanding Polynomial Zeros
Polynomials are mathematical expressions that involve variables raised to whole number powers. The degree of a polynomial is defined as the highest power of the variable in the expression.
Key Concept: The Fundamental Theorem of Algebra
- This theorem states that a polynomial of degree n has exactly n roots (or zeros) in the complex number system, counting multiplicities.
Option Analysis
- Option A: Only one zero
- This is not always true; a polynomial can have multiple zeros or none.
- Option B: At least n zeros
- While a polynomial has exactly n roots, some could be repeated (counted multiple times), but they don’t guarantee distinct zeros.
- Option C: More than n zeros
- This is incorrect. A polynomial of degree n cannot have more than n roots.
- Option D: At most n zeros
- This is the correct answer. A polynomial can have anywhere from 0 to n zeros. Zeros can be real or complex. For example:
- A quadratic polynomial (degree 2) can have 0, 1, or 2 real roots.
- A cubic polynomial (degree 3) can have 0, 1, 2, or 3 real roots.
Conclusion
In summary, a polynomial of degree n can have at most n zeros, which may include repeated roots and complex solutions. This is a fundamental principle in algebra that helps in understanding the behavior of polynomial functions.
Polynomials are mathematical expressions that involve variables raised to whole number powers. The degree of a polynomial is defined as the highest power of the variable in the expression.
Key Concept: The Fundamental Theorem of Algebra
- This theorem states that a polynomial of degree n has exactly n roots (or zeros) in the complex number system, counting multiplicities.
Option Analysis
- Option A: Only one zero
- This is not always true; a polynomial can have multiple zeros or none.
- Option B: At least n zeros
- While a polynomial has exactly n roots, some could be repeated (counted multiple times), but they don’t guarantee distinct zeros.
- Option C: More than n zeros
- This is incorrect. A polynomial of degree n cannot have more than n roots.
- Option D: At most n zeros
- This is the correct answer. A polynomial can have anywhere from 0 to n zeros. Zeros can be real or complex. For example:
- A quadratic polynomial (degree 2) can have 0, 1, or 2 real roots.
- A cubic polynomial (degree 3) can have 0, 1, 2, or 3 real roots.
Conclusion
In summary, a polynomial of degree n can have at most n zeros, which may include repeated roots and complex solutions. This is a fundamental principle in algebra that helps in understanding the behavior of polynomial functions.
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A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer?.
A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A polynomial of degree n has:a)Only one zerob)At least n zeroesc)More than n zeroesd)At most n zeroesCorrect answer is option 'D'. Can you explain this answer?.
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